7 Metacognitive Pedagogies in Mathematics Classrooms
From Kindergarten to College and Beyond
Zemira R. Mevarech, Lieven Verschaffel, and Erik De Corte Theoretical Background
It is well known that the cognitive system depends on higher-order processes that enable it to work efficiently.
Since ‘meta’ means ‘beyond’, those higher-order processes are termed ‘meta-cognition’—they are ‘beyond’ the cognitive system. The main meta-cognitive components include planning, monitoring, control, and reflection (e.g., Flavell, Miller, & Miller, 2002).
A good metaphor for the way metacognition operates is the GPS (Global Positioning System). The GPS, also known as Navstar, is a global navigation satellite system that provides information on cars’ locations, road conditions, and driving time. The GPS chooses the best route to go: it plans the route, monitors, controls, and reflects on the driving until the driver reaches the final destination. When an error occurs, the GPS announces it and recalculates the route. The GPS can also alert the driver on obstacles that are on the way, and sometimes suggests how to bypass them. Obviously, the GPS is not needed when the route is familiar to the driver. Similar to the GPS, metacognition also comprises planning, monitoring, control, and reflection processes. It is particularly essential in solving complex, unfamiliar, or non-routine (CUN) problems, but less (or not at all) necessary when the problem is very familiar and can be solved automatically (Mevarech & Kramarski, 2014).
While there is a large consensus that metacognition has a crucial role in activating and regulating the cognitive system, the term itself includes a large number of different processes that sometimes are overlapping or not clearly defined. Furthermore, over the years the concept itself was enlarged so much that currently it includes almost
every process that relates to overseeing and activation of the cognitive system. From the very beginning, Brown (1987) and later on Schraw and Dannison (1994) distinguished between two components of metacognition:
knowledge of cognition and regulation of cognition. Flavell, who coined the term ‘metacognition’ in 1979, clarified in 2002 that metacognition includes not only metacognitive knowledge (e.g., knowledge about the task, strategies appropriate for solving the task, and personal characteristics relevant to the task), but also metacognitive skills, such as monitoring and reflection (Flavell et al., 2002). Kuhn in 2000 talked about meta-strategic knowledge relating to knowledge about the what, when, how, and why of using strategies for solving a problem.
(The reciprocal relations between cognition and metacognition within self-regulated learning (SRL) are widely discussed by Winne, 2018/this volume.)
Efklides (2006) took another approach. She includes in the metacognitive system also knowledge and regulation of affect (emotions, attitudes, motivation, etc.) which she terms meta-experience (see also Efklides, Schwartz, &
Brown, 2018/this volume). Although there is no question that the affective and cognitive systems work hand in hand during learning and problem solving, some theoretical approaches do not consider this meta-experience system as part of the metacognitive system.
It is not surprising that the wide definition of metacognition has raised various questions, many of which are still open. First, there is much confusion regarding the differences between metacognition and SRL. According to Flavell et al. (2002) and many others (see for example the review by Mevarech & Kramarski, 2014) metacognition is a ‘meta’ concept that includes under its umbrella self-regulation (primarily referring to the metacognitive control functions), meta-strategies, metacognitive knowledge, etc. Zimmerman and Schunk (2011) include under SR cognition, meta-cognition, motivation, affect, and behavior. De Corte, Mason, Depaepe, and Verschaffel (2011) combined the two terms by clarifying the components of adaptive competence in mathematics: the ability to apply meaningful learned mathematical knowledge and skills flexibly and creatively in a variety of contexts.
According to De Corte et al. (2011) these components are, besides (a) meta-knowledge and (b) self-regulation skills, (c) a well-organized and flexible, accessible domain-specific knowledge base, (d) heuristic strategies for problem solving, and (e) positive mathematics-related affects involving attitudes, emotions, and beliefs. The meta-knowledge component refers to knowledge about one’s cognitive functioning (metacognitive knowledge), as well as knowledge about one’s motivation and emotions. Self-regulatory skills embrace skills relating to the self-regulation of one’s cognitive processes (metacognitive skills or cognitive self-regulation), as well as skills for regulating one’s motivational and emotional processes (meta-volitional skills or volitional self-regulation).
Given the above discussion, in this chapter the two terms, metacognition and SR, are used interchangeably, although this choice does not necessarily reflect consensus in the field.
The second open issue denotes the age at which learners can activate and regulate their cognition. While some researchers argue that metacognitive skills are activated only at the age of 10, research of the past ten years shows that these skills emerge earlier than has mostly been assumed before (De Corte et al. 2011).
The third open issue regards the extent to which metacognition is teachable. Most of the current studies provide hard data showing that at all ages learners who are exposed to metacognitive interventions are able to improve their metacognitive skills which in turn affect their mathematical reasoning (e.g., Schoenfeld, 1992; Mevarech &
Kramarski, 2014). Studies based on meta-analysis, such as those conducted by Dignath and Büttner (2008) and Hattie (2009), clearly indicate that SR skills can be enhanced as a result of explicitly teaching those skills.
These issues are used as the framework for the present chapter that focuses on meta-cognitive pedagogies that have been proven to be successful in enhancing students’ mathematical reasoning. Below is an overview of the chapter:
Metacognition, self-regulation, and mathematical reasoning;
Metacognitive pedagogies and mathematics education;
Research evidence regarding the effects of metacognitive pedagogies on the mathematical reasoning of kindergarten children, students in elementary schools, secondary schools, and higher education;
Developing self-regulation skills for word problem solving in primary and secondary school levels;
Developing self-regulation skills for geometry problem solving;
The effects of metacognitive scaffolding on students in higher education;
Self-regulated mathematics learning in ICT environments;
Future research directions; and
Implications for educational practice.
Metacognition, Self-Regulation, and Mathematical Reasoning
The relationships between metacognition and mathematical reasoning are well documented in the psychological and educational literature (e.g., Schneider & Artelt, 2010). Researchers have indicated that students of all ages, K–12 and adults, who plan, monitor, evaluate, and reflect on their problem-solving processes solve mathematical problems better than those who do not use (or use less often) these activities (e.g., Stillman & Mevarech, 2010).
This phenomenon was observed by using a large variety of off- and on-line measurements, including:
questionnaires, observations, interviews, videos, various brain coding techniques (e.g., Magnetic Resonance Imaging, MRI), think-aloud techniques, and analysis of peers explaining the solutions to one another or working in small groups. The earlier studies referred to metacognition as a whole, whereas more recent studies distinguished between the specific components of metacognition, as explained above. In general, studies reported high positive correlations between metacognition and mathematics reasoning, even after controlling for IQ (Veenman & Spaans, 2005). Interestingly, Veenman (2013) and Van der Stel and Veenman (2014) compared the development of IQ to that of metacognition and found different developmental curves for each variable.
Metacognitive Pedagogies and Mathematics Education
The findings reviewed above showing the positive relationships between metacognition and mathematics reasoning led researchers to look for pedagogies for improving students’ metacognitive thinking, their reading comprehension, problem solving, higher-order thinking skills, or their knowledge and conceptual understanding, based on metacognitive processes (Zohar & Barzilai, 2013).
Over the years, several metacognitive methods have been designed for the area of mathematics learning, some being followed by intensive research (e.g., Stillman & Mevarech, 2010; Mevarech & Kramarski, 2014; Schneider
& Artelt, 2010). Most of these methods were routed in the seminal work of Polya (1957) and Schoenfeld (1985, 1992). Generally, these methods use self-addressed metacognitive questions and share common stages as suggested by IMPROVE (Mevarech & Kramarski, 1997; see also Kramarski, 2008/this volume).
The acronym of IMPROVE represents the involved teaching steps:
Introducing the new materials, concepts, problems, or procedures using metacognitive scaffolding;
Metacognitive self-directed questioning in small groups or individually;
Practicing by employing the metacognitive (MC) questioning;
Reviewing the new materials by teacher and students, using the MC questioning;
Obtaining mastery on higher and lower cognitive processes;
Verifying the acquisition of cognitive and metacognitive skills based on feedback-corrective processes; and
Enrichment and remedial activities.
The core component of the IMPROVE consists in training the students to use four kinds of metacognitive self- directed questions:
Comprehension: What is the problem all about?
Connection: How is the problem at hand similar to or different from problems you have solved in the past? Please explain your reasoning.
Strategies: What strategies are appropriate for solving the problem and why?
Review: Does the solution make sense? Can you solve the problem differently, how? Are you stuck, why?
To demonstrate the effects of metacognitive pedagogies on mathematics reasoning, the following section reviews studies that exemplify the implementation of metacognitive pedagogy in kindergarten, primary and secondary school, and higher education, respectively; it provides research evidence on the impact of the metacognitive pedagogies on the mathematics reasoning of K–12 and higher education students. Finally, the impact of self- regulation (SR) scaffolding in ICT (Information and Communication Technologies) environments is shortly reviewed.
Research Evidence
Implementing metacognitive pedagogy in the kindergarten is not at all self-evident. As mentioned above, studies conducted in the 1980s and 1990s claimed that children younger than 10 years old have limited metacognitive skills because they are in the concrete developmental stage and therefore cannot activate higher-order thinking skills, such as those involved in metacognition. However, in the 2000s, research has started to report other evidence. Veenman, Van Hout-Wolters, and Afflerbach (2006) indicated that children at the ages of 4–5 can estimate the difficulty of a task and have some knowledge of which strategies to use. Whitebread and Coltman (2010) showed that without adult intervention, kindergarten children at the ages of 3–5 spontaneously plan, monitor, control, and reflect on their mathematics activities. Earlier, Mevarech (1995) demonstrated that kindergarten children could activate metacognitive processes when encountered with mathematics tasks. For example, children at this age could identify and explain which information is crucial for solving mathematical problems, and they could also distinguish between mathematics and non-mathematics information provided in word problems.
Based on this research, several intervention studies used metacognitive pedagogies for enhancing kindergarten children’s metacognition and mathematical reasoning (e.g., Ginsburg, Lee, & Boyd, 2008). In these studies, the kindergarten teacher scaffolds children’s thinking by providing metacognitive hints based on IMPROVE and asks the kids to explain their reasoning. For example, Mevarech and Eidini (in preparation) conducted a study in which the kindergarten teacher read aloud an e-book embedded with metacognitive scaffolding (Shamir and Baruch, 2012). The metacognitive questions were modified to fit the child’s age: What does this page tell us? What do you have to do in order to find the answer? Please explain your thinking. Why do you think you have to add/subtract? If the children did not know what to do, the kindergarten teacher went with the children to the previous page and asked them: How did you find the answer here? Then she returned to the next page and repeated the questions. The study indicated that exposing kindergarten children to metacognitive pedagogy highly enhanced their metacognition and mathematical reasoning. The experimental group could better explain their reasoning, used richer mathematical language, and improved their problem-solving skills more than the control groups.
Developing Self-Regulation Skills for Word Problem Solving in Primary and Secondary School Levels
De Corte and Verschaffel (2006) designed an innovative learning environment, ‘Skillfully Solving Context Problems (SSCP)’, for fifth graders’ acquisition of adaptive competence in mathematical problem solving. As mentioned in the first section of this chapter, self-regulatory skills constitute a crucial component of adaptive competence.
The SSCP learning environment focused on cognitive self-regulation skills. It consists of a series of 20 lessons taught over a four-month period, and aimed at the acquisition by the students of a self-regulation strategy for problem solving consisting of five stages:
I build a representation of the problem;
I decide how to approach and solve the problem;
I do the necessary calculations;
I interpret the outcome and formulate an answer; and
I control and evaluate the solution.
A set of eight heuristic strategies was embedded and taught in the first and second stages. For example: draw a picture of the problem situation, or distinguish relevant from irrelevant data. Acquiring this problem-solving strategy involved (1) becoming aware of the different phases of a competent problem-solving process (awareness training), (2) becoming able to monitor and evaluate one’s actions during the different phases of the solution process (self-regulation training), and (3) gaining mastery of the eight heuristic strategies (heuristic strategy training).
To elicit and support in all students constructive, self-regulated, situated, and collaborative learning (De Corte &
Verschaffel, 2006), the environment was designed—in narrow cooperation with the teachers of the four participating classes and their principals—based on the following three pillars that embody those characteristics of productive learning.
A varied set of complex, realistic, and open problems that lend themselves well for the application of the self- regulation skills and the heuristics that were intended to develop in students.
Creating a learning community through the application of a varied set of activating and interactive instructional techniques, namely small group work, whole class discussion, and individual assignments. Throughout the lessons, the teacher encouraged the students to reflect upon their cognitive and self-regulation activities. This support was gradually faded out as students became more competent in solving problems, and consequently regulated more and more their own solution activities.
Establishing an innovative classroom culture through the introduction of new social norms with respect to learning and teaching problem solving. Typical aspects of this classroom culture include: (1) stimulating students to articulate and reflect upon their solution strategies and beliefs about problem solving; (2) discussion about what counts as a good problem, a good response, and a good solution procedure; (3) reconsidering the role of the teacher and the students in the learning community.
The teachers were intensively prepared for supporting the implementation of the learning environment. The effects of the intervention were evaluated using a pretestposttest-retention test design with an experimental group consisting of four fifth-grade classes (n = 86) and a control group of seven comparable classes (n = 146). A wide variety of data-gathering instruments was applied: word-problem-solving tests, a standardized mathematics achievement test, an attitude questionnaire, interviews, and video-registration of some lessons.
The findings indicate (see also De Corte, 2012) that the intervention had a significant and stable positive effect on the experimental pupils’ skills in solving math problems (in comparison with a control group). The positive effect was stronger for the high-ability students, but also the low-ability ones benefited significantly from the intervention. The learning environment had also a significant, albeit small positive effect on students’ pleasure and persistence in solving problems and on their math-related beliefs and attitudes. The results on a math achievement test revealed a signifi-cant transfer effect to other parts of the math curriculum (measurement, geometry): the experimental students performed significantly better on this test than the control group. There was a substantial significant increase in the experimental students’ spontaneous use of heuristic and self-regulation skills (orienting, planning, monitoring, evaluating).
Studies by Mason and Scrivani (2004) and by Panouara, Demetriou, and Gagatsis (2010), in which an SSCP- based learning environment for problem solving was used also with fifth graders, yielded similar major findings.
Altogether these studies show that innovative learning environments in which self-regulation skills for solving math problems are learned by using interactive instructional methods in a new classroom culture can significantly increase students’ competence.
Interestingly, the basic principles underlying the interventions applied in those studies converge with the characteristics of the effective learning environments that derive from recent meta-analyses of teaching experiments: (1) train in an integrated way cognitive, metacognitive, and motivational strategies, using thereby a variety of teaching methods; (2) pay explicit attention to the usefulness and benefits of strategies; (3) create opportunities for practicing strategies and provide feedback about strategy use; (4) create an innovative classroom culture that stimulates SRL, especially reflection (Dignath & Büttner, 2008; Dignath, Büttner, & Langfeldt, 2008;
Veenman et al., 2006).
Studies on the effects of metacognitive pedagogy on secondary school mathematics achievement reveal similar findings to those conducted at the lower levels of education. The positive effects were evident not only on
‘traditional’ mathematics achievement tests, but also on math literacy which is largely emphasized in recent years.
PISA (Programme for International Student Assessment of the OECD) defines mathematical literacy as:
The capacity to identify, understand and engage in mathematics as well as to make well-founded judgments about the role that mathematics plays in an individual’s current and future life as a constructive, concerned, and reflective citizen.
(OECD, 2003, p. 20)
Metacognitive pedagogy is particularly beneficial for promoting students’ mathematical literacy because it trains students to activate higher-order cognitive skills which are crucial for solving math literacy tasks. Research findings indicate that tenth graders who solved math literacy tasks via IMPROVE significantly outperformed their counterparts who solved the same literacy tasks for the same duration of time without the metacognitive prompts.
Interestingly, fine-tuning analysis of students’ performance on the math literacy test indicated that the effect size was larger on the ‘application’ compared to the ‘computation’ components (Mevarech & Lianghuo, in press).
To conclude, two meta-analyses (Dignath & Büttner, 2008) based on 49 studies at the primary school level and 35 at the secondary school level that analyzed the effects of SRL on reading and mathematics achievement reported an average effect size of 0.69. For both school levels, higher effect sizes were observed when the training was conducted by researchers instead of regular teachers. Moreover, higher effects were attained in the scope of mathematics than in reading/writing or other subjects. The main conclusion of these meta-analyses was that SRL can be fostered effectively at both primary and secondary school levels.
Developing Self-Regulation Skills for Geometry Problem Solving
Although geometry is an integral part of the mathematics curriculum, surprisingly only a few studies explored the relationships between metacognition and achievement in geometry or the effects of metacognitive pedagogy on students’ achievement in geometry. Kai-Lin (2012) explored the extent to which the use of metacognitive strategies that relate mainly to reading comprehension affects students’ comprehension of geometric proofs. He found that good comprehenders tended to employ more meta-cognitive reading strategies for planning and monitoring and more cognitive reading strategies for elaborating proof compared with moderate comprehenders who in turn employed these strategies more often compared with poor comprehenders.
While Kai-Lin (2012) explored the relationships between cognitive and meta-cognitive reading strategies on comprehension of geometry proof, Mevarech, Gold, Gitelman, and Gal-Fogel (2013) examined the immediate and delayed effects of meta-cognitive scaffolding implemented via IMPROVE on eighth-grade students’
achievement in geometry. In this study, the learning unit was trapezoids: definitions, proofs, and computations.
The study indicates that although prior to the beginning of the study the IMPROVE group scored significantly lower on the pretest than the control, after the intervention the IMPROVE students outperformed the control group on the immediate test that was administered at the end of the intervention, as well as on the delayed posttest administered two months later. Similar findings were reported by Hurme and Järvelä (2005) and Schwonke, Ertelt,