As scientists and engineers drive so much of our innovation and creation of knowledge, high-quality science, technology, engineering, and mathematics education is key to the success of advanced economies. Given its transversal nature, mathematics education is a cornerstone of this agenda.

Education has changed dramatically over recent years from elite education provided to only a small percentage of the population to compulsory education in which no child should be left behind. The skills necessary for the industrial age have been superseded by those deemed appropriate for the knowledge-based world. Our models of learning have also progressed: instead of seeing learners as “tabulae rasae” (“blank slate”) simply absorbing information, we view them as active builders of information, constructing knowledge.

This book is based on tens of studies, all trying to understand how education can foster the skills that are appropriate for innovative societies. It focuses on mathematics education, a subject that is heavily emphasised worldwide, but nevertheless still considered to be a stumbling block for many students. While there is almost a consensus that the mathematics problems appropriate for the 21st century have to be complex, unfamiliar and non-routine (CUN), most of the textbooks still include only routine problems based on the application of ready-made algorithms. The challenge might become even greater as the development of mathematics literacy comes to be one of the key aims in the curriculum. Undoubtedly, there is a need to introduce innovative instructional methods for enhancing mathematics education and in particular students’ ability to solve CUN tasks. These require the application of metacognitive processes, such as planning, monitoring, control, and reflection. It will be critical to train students to “think about their thinking” during learning.

Problem solving is at the core of all mathematics education. The solution of complex, unfamiliar and non-routine (CUN) problems has to be the cornerstone of any effective learning environment for mathematics for the 21st century. While students solving routine problems can rely on memorisation, solving CUN problems requires mathematical skills that include not just logic and deduction but also intuition, number sense and inference. Innovative societies require creativity in mathematics as well as in other domains. The approach to mathematical communication has also changed, with students in all age groups being encouraged to engage in mathematical discourse and share ideas and solutions as well as explaining their own thinking. Developing these competencies may result in enhancing social skills as well as mathematically literate citizens.

The term metacognition was first introduced to indicate the process of “thinking about thinking”. Since then the concept has been elaborated and refined, although the main definition has broadly remained the same. Metacognition is now recognised to have two main components: “knowledge of cognition” (declarative, procedural and conditional knowledge), and the more important “regulation of cognition” (planning, monitoring, control and reflection). Basic metacognitive skills appear to start to develop in very young children and grow in sophistication with age and intellectual development. It is not yet clear how far metacognitive abilities in one domain can be transferred into another, but there is a strong relationship between metacognition and schooling outcomes with implications for educators, researchers and policy makers.

Can metacognition be taught? And if so, what are the conditions that can facilitate metacognitive application in the classroom? While the research shows that metacognition can be successfully taught, implicit guidance is not enough. Co-operative learning should help to foster metacognition by providing ample opportunities for students to articulate their thinking and be involved in mutual reasoning, nevertheless students still have to be taught how to apply these processes and also intensively practise them. Effective metacognitive guidance needs to be explicit, embedded in the subject matter, involve prolonged training, and inform learners of its benefits. A number of methodologies for teaching metacognition have been developed, all of which use social interactions and selfdirected questioning in order to encourage learners to be aware of their metacognitive processes and apply these processes in learning.

This chapter reviews the five main metacognitive pedagogies used in maths education, their benefits and trade-offs. The models are: Polya, Schoenfeld, IMPROVE, Verschaffel and Singapore. All of them use some form of self-directed questions but differ in their details, scope and age range. Polya’s and Schoenfeld’s models are designed to be used with university students and on single CUN problems, whereas IMPROVE, Verschaffel’s model and the Singapore model can be used with younger learners and for a set of problems or even a whole curriculum. IMPROVE has also been modified for use in other domains, and for teachers’ professional development with or with no advanced technologies. Comparing the models highlights the advantages and challenges associated with each one of them.

Understanding the rationale behind a teaching method and accepting the assumptions on which it is based are not enough. Policy makers, educators and even the public at large look for evidence on its effects on the one hand, and on its drawbacks on the other. A large number of experimental and quasi-experimental studies have been carried out into the effects of metacognitive pedagogies on mathematics achievement, always comparing the metacognitive group to a control group that was taught traditionally. Among school children of all ages, metacognitive approaches improve achievement in arithmetic, algebra and geometry, with lasting effects, and positive effects even in high-stakes situations such as matriculation exams. Effects are similar but smaller for college students. Metacognitive approaches were mostly more effective within co-operative settings, although they also improved achievement among individualised settings.

Emotion and cognition are inextricably linked in the brain. Social skills are essential to the process of learning and the evidence shows that metacognitive interventions designed to improve cognitive achievement can have a beneficial impact on affective factors such as motivation or anxiety. In addition, metacognitive methodologies can be adapted to promote social-emotional competencies among kindergarten pupils, primary and secondary school students, and adults. Combining the two approaches has an even greater impact on both social-emotional and cognitive achievements than either one on its own. Interventions that focus only on motivation or only on cognitive-metacognitive competencies are more effective than traditional instruction, but less effective than focusing on both motivation and metacognition.

Information and communications technology (ICT) could be a powerful tool for teaching mathematics, and particularly the solving of CUN tasks, but its potential has not always been fulfilled, possibly because 1) ICT-enhanced learning environments create cognitive overload; 2) meaningful learning with ICT depends on students being able to monitor, control and reflect on their learning; and 3) the type of metacognitive scaffolding provided in these environments needs to be tailored to the characteristics of the individual technologies. This chapter focuses on three kinds of ICT environments embedded within metacognitive pedagogies: specific maths software, general e-communication tools such as asynchronous learning networks, and general software such as e-books. Some of them are still in their infancy but they all appear to benefit from the addition of metacognitive scaffolding whether embedded into the technology itself or provided externally by a teacher.

Teachers and principals have an important role in introducing change in schools. Given that “one cannot teach what one does not know”, teachers’ own metacognitive skills are increasingly being studied. Observations have shown that although teachers seldom explicitly activate metacognitive processes while teaching, they do apply them implicitly in the classroom. Their understanding of metacognition is related not only to their practice, but also to their students’ self-regulated learning and achievement. Professional development programmes are the natural settings for the introduction of innovative teaching methods. Studies into the effects of metacognitive pedagogies on both in-service and pre-service teachers have found they positively enhanced teachers’ knowledge, pedagogical-content knowledge (PCK), self-regulated learning (SRL) and self-efficacy, but these studies have not followed teachers into the classroom.

This chapter summarises the main findings of the book and concludes.