Want to improve your problem-solving skills? Try metacognition

Critical mathsToday’s post is by Anne-Lise Prigent the editor in charge of education publications at OECD Publishing

French poet Paul Valéry once expressed his love for mathematics: “I worship this most beautiful subject of all and I don’t care that my love remains unrequited.” Unrequited love, or, all too often, a big stumbling block that inspires fear and defiance, mathematics are usually not seen as an excuse to have fun. Yet, maths need not go hand in hand with anxiety.

Maths can help us acquire life skills that are essential. In today’s unpredictable world, all of life is problem solving. Our societies will need a critical mass of innovators who can not only communicate and collaborate but also think creatively and critically. Are students developing these high-order skills in maths class? Are they learning to solve complex, unfamiliar and non-routine tasks?

A new OECD publication, Critical Maths for Innovative Societies: The Role of Metacognitive Pedagogies, shows that the time has come to introduce innovative instructional methods. The man behind the project is Stéphan Vincent-Lancrin, who worked with authors Zemira Mevarech and Bracha Kramarski. Faithful to the spirit of CERI (the OECD’s Centre for Educational Research and Innovation), Vincent-Lancrin rooted the report in CERI’s thorough knowledge of learning (The Nature of Learning, Innovative Learning Environments). How do we learn? Not by having cold sweats at the sight of an equation assuredly. Emotions play a key role in learning – our cognitive and emotional systems are intertwined in our brain. But do we ever learn how to learn?

College professors often point out that their students never learnt how to learn. Derek Cabrera was surprised to find that even the “cream of the crop of our education system” was not good at dealing with novel problems in unstructured assignments. As PISA shows, across OECD countries, about one in five students is able to solve only straightforward problems – if any – provided that they refer to familiar situations. Too often, we teach students what to think but not how to think.

Yet, there is an engine we can use for that and it is called metacognition, which means “thinking about your thinking”, and regulating it. Metacognitive pedagogies improve academic achievement: content knowledge and understanding, and the ability to handle routine and unfamiliar problems. And they also boost affective outcomes, reducing anxiety and improving motivation. Struggling students greatly benefit from these pedagogies, but not at the expense of higher achievers.

Metacognition is about taking ownership of your learning and maximising it. “It turns you from being a consumer of learning to being a researcher, a co-producer, an explorer and that’s a much more exciting, exhilarating world. You discover how to learn better” Stephen Heppell argues. He also points out that metacognition makes students “do 20% better – you get an extra Friday every week”.

How does it work? Among metacognitive pedagogies, the IMPROVE method is the most widely studied: Introducing the whole class to the new material, concepts, problems or procedures by modelling the activation of metacognitive processes. Metacognitive self-directed questioning applied in small groups or individualised settings. Practising by employing the metacognitive questioning. Reviewing the new materials by the teacher and the students, using the metacognitive questioning. Obtaining mastery on higher and lower cognitive processes. Verifying the acquisition of cognitive and metacognitive skills based on the use of feedback-corrective processes. Enrichment and remedial activities.

It helps learners build their own scaffolding: “What is the problem all about?”, “Have I solved problems like that before?”, “What strategies can I use?”, “Am I stuck, why? What additional information do I need? Can I solve the problem differently?”. The key is to ask ourselves questions that help us change our perspective and connect the dots in new ways.

In doing this, we inevitably make mistakes. And that’s good. Mistakes are data that help us perform better later. And co-operative learning makes these pedagogies even more efficient. Students using IMPROVE interact, share rich explanations, evaluate their peers’ solutions, correct mistakes, and learn. Communication in control groups may be dull – or altogether absent, but groups using IMPROVE discover a new, friendlier way of learning maths. They learn to learn and think creatively.

Now, are these pedagogies used in classrooms today? Not so much. Singapore – who excelled in PISA and TIMSS – is an interesting exception. Teachers there are trained to use metacognitive pedagogies explicitly in class. Singapore adopted metacognition in its mathematics curriculum for all school grade levels at the start of the 2000s.

From kindergarten to higher education, metacognitive pedagogies are effective across all levels. And they can easily be used in other domains (e.g. science) since the metacognitive questioning is generic. As Stéphan Vincent-Lancrin points out in the podcast below, there is a message for policy makers here. “If you want (…) less anxiety, more communication, a better understanding of maths, better reasoning etc., there are some interesting pedagogies which have already been tested.”

Metacognitive pedagogies are particularly effective for complex, unfamiliar and non-routine tasks, but they also work for routine tasks. However, it is time maths textbooks caught the 21st century train: 90% of the problems featured in these textbooks are routine problems.

It would also be interesting to explore whether metacognitive pedagogies could help decrease the gender gap. PISA 2012 revealed worrying gender differences in students’ attitudes towards mathematics. Girls report less openness to problem solving than boys, less perseverance, less belief in their own skills, and higher levels of maths anxiety – even when they perform as well as boys in maths.

Those who develop metacognitive skills will use them all their lives. As Howard Gardner puts it, “Those with flexible minds, with open minds, are at a distinct advantage overall. (…) So are those who know how their own mind works and can marshal that metacognition knowledge in cases where the course to pursue is not clear (…) The search of truth going forward must become ever more metacognitive. We can no longer trust our eyes or the spoken words of the nightly news (…) we must try to understand the truths about truth.”

A good example of this is the astonishing story of “the British amateur who debunked the mathematics of happiness”. Nick Brown, a first-term, first-year, part-time master’s student in his 50s, managed to prove America’s academic establishment wrong. How did he do that? He used metacognition: “the maths you need to understand the Losada system is hard but the maths you need to understand that this cannot possibly be true is relatively straightforward”, he argues. In the same way, the Wright brothers – two bicycle mechanics – invented the first airplane when they understood that aviation pioneer Lilienthal had got his maths wrong – although the scientific world had accepted his findings.

A sound judgment and perseverance. Resilience. As you walk through the valley of maths, have no fear. The grace of an algorithm can go a long way. And as you ponder the beauty of space geometry, mark Leibniz’s words: “Music is the pleasure that human minds experience with counting without realizing that they are counting” (Marcus du Sautoy). Counting on, Leibniz invented the calculus (not that Newton would agree, as both – curious – minds seem to have converged in their intellectual prowess). Moving on, let your metacognitive strides swing you past some pretty hefty hurdles – melodiously, mathematically.

Critical maths: entretien avec Stéphan Vincent-Lancrin en français

Critical maths: interview with Stéphan Vincent-Lancrin

Useful links

Maths education for innovative societies by Stéphan Vincent-Lancrin on the OECD educationtoday blog

PISA 2012 Results: Creative Problem Solving (Volume V) – Students’ Skills in Tackling Real-Life Problems

Intuition and ingenuity: Alan Turing’s work and impact

Don't blame Alan Turing
Don’t blame Alan Turing

Legend has it that Apple’s rainbow-coloured logo showing the apple with a bite out of it is in homage to Alan Turing “the father of modern computing”. Turing died of cyanide poisoning on 7 June 1954, two years after being convicted of homosexuality and accepting chemical castration instead of prison. A half-eaten apple was found next to him, and one theory is that he’d laced it with cyanide, his own homage to the wicked queen in Snow White, his favourite Disney cartoon. Another theory is that he died accidentally after inhaling cyanide fumes from apparatus he had in his bedroom for electroplating spoons. A third explanation is that he really did commit suicide, but set up the apparatus so his mother would think it was an accident. The coroner didn’t test the apple for cyanide, so we’ll never know for sure.

If there are doubts about Turing’s death, his life is fairly well-known, or at least some aspects of it. His most noteworthy exploit for the general public was helping to break the code of the Enigma machines the Germans used to communicate with their submarines during the Second World War. If you’d like to get some idea of how he did it, take a look at the excerpts from the “Enigma Paper” in Alan Turing, His Work and Impact, just published by Elsevier. Cryptography is the second of four parts of this thousand-page overview presenting Turing’s most significant works from the four-volume Collected Works along with comment, analysis and anecdote from leading scholars. The other three parts are on Turing’s contributions to computability, artificial intelligence, and biology.

That simple naming of the parts already gives you some idea of the breadth of Turing’s influence, and we could also add economics. I actually got the Elsevier book thanks to Professor K. Vela Velupillai who wrote for us about Turing’s economics here.  That article described the foundations of computable economics, while here at the OECD the project on new approaches to economic challenges was being launched. “New Approaches” revisits some of the fundamental assumptions about the functioning of the economy, and the implications for policy. It also addresses how to extend the capabilities of existing tools for structural analysis and analysing trends over the long term to factor in key linkages and feedback – for example between growth, inequality, and the environment.

Vela cites Turing’s Precept, an idea that should be kept in mind by economic theorists, analysts and policy makers everywhere: “the inadequacy of reason unsupported by common sense”. There’s a corollary to that in how you present the reasoning, best summed up by the German mathematician David Hilbert at the 1900 conference of the International Congress of Mathematicians in Paris. Presenting a paper on 23 unsolved problems that would help set the research agenda for mathematics in the new century, Hilbert quoted an old French mathematician as saying: “A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street”. And Turing himself claimed that “No mathematical method can be useful for any problem if it involves much calculation.”

Gregory Chaitin recalls Hilbert’s remark when he presents Turing’s “Solvable and unsolvable problems”, that ends with Turing’s Precept. In this “lovely paper” Turing explains the notion of computability and proves the unsolvability of a decision problem without using any mathematical formalism. His “models” are two puzzles that were popular at the time: a picture made up of a number of movable squares set in a frame, with one square missing so you can move the squares around to form the image; and two pieces of intertwined wire you can separate without bending or breaking them.

That said, much of what’s presented is for specialists and you’d need a good grounding in mathematics to follow it. But there’s still plenty even for a non-mathematician like me, some of it surprisingly moving, for example when Bernard Richards describes how he presented his and Turing’s work on morphogenesis to Turing’s mother shortly after his death. Some of it is intriguing – why does the UK government still refuse to declassify the two 1946 papers “Report on the applications of probability to cryptography” and “Paper on statistics of repetitions”? But no matter how well you know the life and work of Turing, you’ll learn something from this book.

By the way, that Apple story at the beginning is only a legend. Rob Janoff who designed the logo explained that he was asked to come up with something simpler than the (hideous) picture of Newton sitting under an apple tree that was the company’s first logo, and the bite was just to show that it was an apple, not a cherry or a tomato. The gay-friendly rainbow was to advertise the colour graphics capabilities of Apple’s computers. On British TV show QI XL, Stephen Fry recalled asking  his friend Steve Jobs about the Turing story “It isn’t true, but God we wish it were!” was Jobs reply.

Useful links

Models used in the OECD Economics Department