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

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