Learning to solve problems


“Robots – memorising formulas, regurgitating facts. Pretending that we understood when we didn’t have a clue. Playing by the rules of the examiner. That’s before we met Mr G, who rescued us from the education system that failed us all. He was a teacher who relished making you think, and not telling you the answer. The one whose enthusiasm was contagious.”

This quote is from last year’s ‘My Education’ report. Could a teacher hope for higher praise from a pupil? She says that because of her teacher, she has discovered the beauty of mathematics and learned how to think.

Thinking is not easy. Teaching pupils how to think is much harder than getting them to remember things. “People are naturally curious, but we are not naturally good thinkers; unless the cognitive conditions are right, we will avoid thinking.” This is one of the main findings from cognitive science, as summarised by Daniel Willingham in his book “Why Don’t Students Like School?”

Cognitive scientists have tested this with numerous experiments, concluding that the brain is far better suited to support the ability to see and move, as opposed to think. For example, consider this paradox. “A teacher tells the students that there will be an unexpected test next week, but the test will be a surprise. They will not know the day of the test until it actually happens.” Can you figure out when the test will be?

I was asked this many years ago, when I was being interviewed for a place to study at university. The interviewer, a charming, eccentric man – chain-smoked through the interview, and was constantly drinking this strong, smoky tea – encouraged me to think, step-by-step. Could the test be on Friday? Why not? If not Friday, then how about Thursday? With some guiding questions and encouragement, I eventually managed to solve the paradox. (Click here to read more about this paradox.)

The value of problem solving

But if teaching how to think is hard, is it worth the effort? Many educators will probably agree that the ability to think is valuable in itself. Thinking will enable us fulfil our unique human potential. However, beyond this intrinsic value, thinking is also valuable in practical terms.

Consider the chart below: the number of jobs in the US economy that require working collaboratively to solve non-routine problems grew dramatically from 1960 to 2000. (Source: The Learning Society report by Cisco.)

Nonroutine tasks.jpg

In the original research paper, a task was defined as routine if “it can be accomplished by machines following explicit programmed rules”. For example, moving a windshield into place on an assembly line, and many other tasks where you do the same thing over and over again. As is evident form the graph, these tasks have been declining most rapidly since the 1980s. In contrast, nonroutine interactive tasks have been growing quickly. This is where you have to solve new problems, while interacting with other people.

There is nothing radically new about the so-called 21st century skills such as critical thinking, collaboration, creativity etc. However, the reality is that far more people, compared to just 20-30 years ago, need to master these skills to earn a living.

McKinsey & Co recently conducted a major survey across Europe. The idea was to find out which skills are valued by employers but missing among young people. In most countries surveyed, the picture was quite similar. Pretty consistently, among the top four missing skills were: problem solving and analysis, teamwork, spoken communication, and work ethic.

Somewhat surprisingly, lack of skills was perceived as a less urgent issue in the United Kingdom compared to Germany, France and other European countries. Yet, just a few months ago, there were 940,000 unemployed 16 to 24 year olds in the United Kingdom. Youth unemployment rate was 21 percent. Young people need better skills (in the UK, especially vocational skills) to match the needs of employers. This is a very complex problem to solve – which in itself is further evidence about the importance of collaborative problem solving.

Problem solving in mathematics

How can one learn to solve problems like the one I was asked in my university interview? Let us start by exploring how problem solving can be taught and learned in the context of one subject, mathematics, and then look at other subjects. The following recommendations are based on a practice guide published by What Works Clearinghouse in the US. After careful review of numerous research papers, the panel of experts put together a list of recommendations to teach problem solving in mathematics in grades 4-8.

Recommendation 1: Prepare problems and use them in whole-class instruction

The idea here is to find both routine and non-routine problems for students to solve. Non-routine meaning problems for “which there is not a predictable, well-rehearsed approach or pathway explicitly suggested by the task, task instructions, or a worked-out example”. A couple of useful links: sample problems can be found from the Illuminations site, the Math Forum, practice problems from PISA etc. When selecting which problems to use, it is important to ensure that students will understand the problem. If they don’t know the context or language, then their problem solving capacity is taken up by trying to understand what is meant by the question. Teachers can anticipate these issues, and select problems with familiar contexts. Also, it may be helpful to reword problems, using words that are connected to pupils’ previous experiences.

It is also helpful to consider students’ previous knowledge of mathematical content when selecting problem-solving tasks. Problems aligned with the current unit often require skills taught in previous years. It may be useful to review skills learned earlier, which are needed to solve the problem. Struggling students are likely to find it especially useful.

Recommendation 2: Assist students in monitoring and reflecting on the problem-solving process.

Another useful strategy is to provide students with a list of prompts that help them think during problem solving. They can be in the form of questions or simple tasks lists. (See a couple of examples below, from the same practice guide.)

Problem solving prompts.jpg

There are a few different ways in which prompts can be shared with pupils. They can be posted on the board, included on worksheets, listed on index cards. In addition, teachers can play the helpful role of modelling (thinking aloud) how to monitor and reflect on the problem-solving process.

Recommendation 3: Teach students how to use visual representations.

This is another simple technique with robust research evidence. Selecting (appropriate) visual representations is likely to be very helpful. For example, schematic diagrams are useful for ratio and proportion problems, percent bars for percent problems, strip diagrams for comparison and fraction problems etc.

Here is an example from the same practice guide. “There are 4 adults and 2 children who need to cross the river. A small boat is available that can hold either 1 adult or 1 or 2 small children. Everyone can row the boat. How many one-way trips does it take for all of them to cross the river?”

I quite like this first visual representation.

Problem solving river1.jpg

It’s a nice little drawing, but the only issue is that it lacks relevant details for actually solving the problem and it includes some irrelevant details. This next one does a better job.

Problem solving river2.jpg

Recommendation 4: Expose students to multiple problem-solving strategies.

Evidence suggests that if you know how to use multiple strategies, you are likely to be more successful. That’s why it is important for teachers to provide instruction in multiple strategies, sometimes even using unsuccessful strategies. This will enable pupils understand that in some situations one needs to try more than one approach to solve a problem. Providing students with worked examples so that they can compare multiple strategies next to each other is another useful practice. This is an important takeaway: research has shown that studying worked examples is a time-efficient way of learning multiple problem-solving strategies.

Recommendation 5: Help students recognise and articulate mathematical concepts and notation.

Mathematical concepts and notation, once pupils are comfortable with them, will help them think about the problem. As always, one should pay attention to pupils’ prior knowledge of concepts and notation, and start from there. When observing the way pupils are solving the problem, teachers can look for opportunities to call out when they use mathematical concepts and notation. Another idea is to use small-group activities so that pupils can discuss the process how they had solved a problem in a worked example, and importantly, the reasoning behind each step.

The full practice guide on mathematical problem solving includes more detailed guidance, along with numerous examples, and ways to overcome common roadblocks in implementing these ideas. This very helpful guide can be downloaded here. Besides mathematics resources, What Works Clearinghouse also includes practice guides and helpful reports and reviews on many other subjects.

Transferring the skill of problem-solving

As we have seen, research has some helpful suggestions how to develop problem solving in mathematics. But is this skill transferable? If students become proficient in mathematical problem solving, will they be able to solve problems in other subjects?

Transferring problem-solving skills to different domains is difficult. In a wonderful article (“Critical thinking: Why Is It So Hard to Teach?”), Daniel Willingham reviews evidence about the impact of various critical thinking programmes and suggests some reasons why their success has been limited. Critical thinking and problem solving are not general skills that can be applied to any situation, after they have been learned. “The processes of thinking are intertwined with the content of thought (that is, domain knowledge). Thus, if you remind a student to “look at an issue from multiple perspectives” often enough, he will learn that he ought to do so, but if he doesn’t know much about an issue, he can’t think about it from multiple perspectives.” Willingham cites that programmes including puzzles like those found on IQ tests report gains in IQ scores, but no significant gains in solving other types of problems.

Does this mean that problem-solving and critical thinking should not be taught? Well, they can be taught and learned, but not easily. The fact that more than 55 percent of students in Shanghai who took part in PISA 2012 were judged to be highly proficient in mathematical problem solving should give us encouragement. This compares to 12 percent of students across OECD and 13 percent of students in the UK who are able to reach this level.

As Willingham concludes in the article mentioned above, there are thinking strategies that, once learned, make critical thinking more likely. This does not mean that the ability to solve problems or think critically will then automatically translate to other domains. It does only if one has sufficient knowledge in the other domain and sufficient practice in using these thinking skills with different types of problems.

Helping your pupils become critical thinkers and problems-solvers is a worthwhile aim. With a lot of attention to domain knowledge and smart, diligent practice over a long period of time, it should be an achievable goal.

If dogs can learn it, then human beings can too. This is an actual advert I saw in the park 🙂

Problem Solving For Dogs.jpg

Better start now

This happened about two weeks ago. I get a phone call from my wife a few minutes after she left home on her bike. “I was hit by a car. The guy just didn’t see me.” My heart sank.

This was the third time in the past couple of years that someone very close to me had been in a cycling accident. Luckily, my wife, my sister and my best friend all survived and are doing fine.

Moments like this have reminded me: whatever I really want to do in life, I better start now.

I left my job at Teach For All in December. In many ways, I had the best job you could imagine. Together with an awesome team, we were supporting the professional development of teachers and school leaders around the world. We were making a difference, and we had fun. My colleagues were some of the nicest, smartest and most hard-working people I’ve ever met. Plus, I was able to travel the world, visit some pretty incredible schools in about a dozen countries. A couple of months ago, we were in Shanghai, the world’s top-performing school system. I’ve been to some really innovative schools, in places like the US and India, showing today what the future of education will look like. As well as many amazing schools up and down the country here in England. I think I learned more in my previous job than I could have done anywhere else.

(Here are some of the schools I have visited in the past 5 years.)


So why leave a fantastic job and jump into the unknown, by starting a new education charity? Is it because of the freedom that being an entrepreneur gives you (being your own boss!), or the hope of making a mark on the world, or making more money? (Just for the record, I have taken a big pay cut and don’t yet have a secure salary for this year.)

I think most importantly, it’s about seeing a need and feeling that something can be done about it.

Every morning I hear news about South Sudan, or youth unemployment in Europe, or the popularity of Golden Dawn in Greece. Which makes me wonder: how will our complex global and local challenges be solved if the next generation is not smarter and more creative than we are?

I’m thinking about all of my friends in education who are working really hard, many of them making huge personal sacrifices. They’re making a difference, no doubt about it. But despite their best efforts, the challenge of improving the quality of education is still huge. Across the OECD, more than one in five 15-year-old students have not mastered basic skills in mathematics. Fewer than one in ten can understand and critically evaluate an unfamiliar text (PISA 2012).

So what can I do to support schools and help make learning more effective? What would help improve academic achievement while ensuring that students are learning the knowledge, skills and personal attributes they most need for life? Can this be done without expecting a super-human effort from teachers and school leaders?

There are no magical solutions. However, as Professor Robert Coe explains in his excellent essay (text, video), high quality professional development for teachers can make a real difference on student learning. A CUREE / Pearson report on research evidence highlights the sort of professional development likely to improve student outcomes: it needs to be collaborative, supported by specialist expertise, focused on aspirations for students, sustained over time and exploring evidence from trying new things.

At Creative Generation, we will be piloting a new, intensive professional development programme for teachers. 10 days per year, 4 teachers per school. Engaging with education research, trying out new practices, evaluating how the new teaching approaches have an impact on student learning, reflecting on what works (or doesn’t) and why. Over time, supporting the professional development of colleagues in your school.

I decided to leave an amazing job and start a new education charity to tackle a really complex challenge. And, of course, because of the prize. Seeing students fulfil their potential and become positive contributors to the world. Seeing a new creative generation emerge.

On excellent teaching and human excellence

Just a minute

Spending a week in Mumbai during the Synergies workshop and a day in Pune – the other city where Teach For India places its fellows – helped me clarify one of the big questions I’ve been asking myself for a long time.

Four years ago we were setting up Noored Kooli, the Estonian programme within Teach For All. Among the group of founders, none of us had been teachers. Once we had recruited some of our country’s brightest young people into the programme, we were struggling with the question – what do we expect them to do in the classrooms? If we want them to be excellent teachers, then what do we really mean by excellent teaching? We just did not know…

I still don’t know, but being part of the Teach For All community, I’m sure we’re getting closer to having an answer to this question. A couple of experiences in Mumbai and Pune helped me clarify my own thinking on this.

Aditya Natraj, founder of the Gandhi fellowship, facilitated a discussion on personal transformation. The key theme that connected various aspects of the talk was ‘human excellence’. One of the definitions really stuck with me- human excellence means becoming the best that I can be. When I was started to think about this in the context of teaching, the obvious question was – what does this mean for a teacher? Does it not mean that we should help all of our students become the best that they can be? If we wanted to do just that, what exactly should we do?

It seems a big part of the answer has to do with ensuring that every child learns to the very best of their ability. In almost all the classes that I have visited recently there are some students – and sometimes a lot of them – who have fallen behind their peers in terms of academic achievement. Unless there is a teacher who inspires them to be the best that they can be, many of them will have very few opportunities in their future, both educational and job opportunities. Unless there is a teacher who believes in them, many will continue to have low aspirations and therefore will never become the best that they could be.

Now that I’ve returned to our office in London, I’ve been thinking about how some Teach For India teachers were helping their students become a little bit better every day. I felt that some of the core values I saw in the classrooms in Mumbai and Pune were very helpful for this. For example:

Work hard.
Be nice.
Try. Try. Try.
Think and question.

And one of my favourite ideas I came across in a Mumbai classroom: Grace under pressure.

Above is a picture of a student who spoke no English a few months ago, and now had to speak for one minute – in English – on a topic of his own choice. Last time he had the same task a few weeks ago, he did not have the courage to say a single sentence, and tears came to his eyes. This time I witnessed him speaking for about 30 seconds and then bravely standing there until the minute was over. When the minute was over, his teachergave him a huge celebration to congratulate him on the progress he had made over the past few weeks! And you could see how happy he was about it!

I was in this classroom thinking that if students become comfortable in situations where they feel a lot of pressure, this will be hugely valuable to them in their future lives. That will help them face difficulties, either at school, at work or at home, and have the courage to overcome them.

The incredible thing was that students in this class had made, on average, 2 years of academic progress in the past 7 months! So clearly, it was a class that was focused on values and achievement. And clearly there was a teacher that inspired his students to be the best that they can be.

I’d love to hear reactions from everyone – what do you think is the role of human excellence in teaching? What are the challenges and joys of helping students become the best that they can be?