The Role and Significance of Metacognition in Teaching Mathematics - Essay Example

?The role of meta-cognition in teaching Mathematics to the International Baccalaureate Primary Year Program learners Mathematics is such an integral part of human life that nobody’s day ever gets completed without having done mathematics in one way or the other. As each day goes by, we do mathematics both consciously and unconsciously. Right from cockcrow to sunset, one is likely to read the time, adjust the clock, but an item, sort out objects, transact business, read the calendar, write a date, check the speedometer, change TV channel and so several other activities that involve mathematics both directly and indirectly. It is for this importance that mathematics holds in our everyday life that the approach towards the teaching of mathematics in school must be done with so much circumspection. Because mathematics is part of our everyday life, it must be taught in such a way that students will adopt concepts through relational learning rather than though rote learning. According to Fox (2009), relational learning has taken place when teachers realize that “learning isn’t just an academic exercise designed to score individuals on their ability to regurgitate information. Rather, it is a lifelong process of understanding truth, gaining wisdom, and making better life decisions” and therefore approaches teaching with methods that are interactive and practical. This is particularly important to ensure at the basic level such as the International Baccalaureate Primary Years. This is because at the primary level, students’ understanding of what they learn is dependent upon relating ideas to their own experience (Junior Achievement Michiana, 2007). One educational concept that plays major role when talking about relational or practical learning of mathematics is meta-cognition. Key words: Cognition, Metacognition. The term Cognition and Metacognition Cherry (2011) defines cognition as “the mental processes involved in gaining knowledge and comprehension, including thinking, knowing, remembering, judging and problem-solving.” Metacognition refers to one's knowledge concerning one's own cognitive processes or anything related to them, e.g. the learning-of relevant properties of information or data.(Flavell, 1976, p. 232). This means that Metacognitive knowledge can be described as the knowledge, awareness, and deeper understanding of one’s own cognitive processes and products (Flavell 1976). Metacognitive skills can be seen as the voluntary control people have over their own cognitive processes (Brown 1987). This transformation suggests changes both in curricular content and instructional style. It involves renewed effort to focus on: • Seeking solutions, not just memorizing procedures; • Exploring patterns, not just memorizing formulas; • Formulating conjectures, not just doing exercises. As teaching begins to reflect these emphases, students will have opportunities to study as an exploratory, dynamic, evolving discipline rather than as a rigid, absolute, closed body of laws to be memorized. For instance in Mathematics: When we solve the sum or a problem we are using ‘Cognition’, that is we are forced to think of different strategies to solve the problem and ‘Metacognition‘ is when we cross-check the answer, maybe we could scrutinize each and every alternative in a multiple-choice task before deciding which is the best one. According to Lucangeli et al (1995), since Flavell introduced the concept of metacognition in 1976, most authors agree that the construct can be differentiated into a knowledge and skills component. It has long been assumed that metacognition—thinking about one’s own thoughts—is a uniquely human ability. Yet a decade of research suggests that, like humans, other animals can differentiate between what they know and what they do not know. They opt out of difficult trials; they avoid tests they are unlikely to answer correctly; and they make riskier ‘‘bets’’ when their memories are accurate than they do when their memories are inaccurate. Anyone who has tried to remember the name of a long-forgotten friend or well-known public figure is probably familiar with the ‘‘tip of the tongue’’ experience—that frustrating feeling that, even without recalling a word or name, you know that you know it (Schwartz, 2002). Knowing that you know involves metacognition, the ability to think about one’s own thoughts, make judgments about one’s own memories, and generally (as the name suggests) engage in cognitive processing about one’s own cognition. Metacognition is omnipresent in daily life, though unlike the tip-of-the-tongue experience, it often goes unnoticed. It can be as simple as conveying a feeling of uncertainty. S-R theory (Behaviorist theory) The stimulus-response theory is justified by the use of drills. According to Thorndike (1922), Drills are means of creating and increasing the stimulus-response bonds that are seen as constituting the S-R theory of arithmetic. The stimulus-response theory is built from Thorndike’s books published in 1922 called the psychology of arithmetic in which he introduced the law of effect, law of readiness and law of practice. Over the years however, other learning theorists have challenged the use of drill as the sole principal method of classroom instruction. To such who oppose, their argument is that drills does not cater for mathematics as an integrated set of patterns and principles but sees mathematics as a group of secluded bonds. This way, they believe drill does not enhance quantitative thinking. In1935 William Brownell objected to the drill method and bond theory of learning. He focused on the use of variety of procedures rather than restricting students to memorize concepts and recall them during drills Brownell emphasized meaningful learning theory and argued: “first bond theory takes no account of quantitative differences in the computation of children; second the drill method implied a distorted view of the goal of learning (Brownell, 1935).” In the view of Brownell and those who accept his school of thought, memorizing to respond 100% accurately to arithmetic questions should not be prime aim of the teacher. Rather, teachers especially those at the primary and other basic levels should focus attention on students’ ability to think quantitatively and with precise understanding of concepts. In the view of Brownell (1928), Learning is the result of associations forming between stimuli and responses. Such associations or "habits" become strengthened or weakened by the nature and frequency of the S-R pairings. The paradigm for S-R theory was trial and error learning in which certain responses come to dominate others due to rewards. The hallmark of connectionism (like all behavioural theory) was that learning could be adequately explained without referring to any unobservable internal states. Connectionism was meant to be a general theory of learning for animals and humans. Thorndike was especially interested in the application of his theory to education including mathematics (Thorndike, 1922), spelling and reading (Thorndike, 1921), measurement of intelligence (Thorndike et al., 1927) and adult learning (Thorndike at al., 1928). For example: The classic example of Thorndike's S-R theory was a cat learning to escape from a "puzzle box" by pressing a lever inside the box. After much trial and error behavior, the cat learns to associate pressing the lever (S) with opening the door (R). This S-R connection is established because it results in a satisfying state of affairs (escape from the box). The law of exercise specifies that the connection was established because the S-R pairing occurred many times (the law of effect) and was rewarded (law of effect) as well as forming a single sequence (law of readiness). Reinforcement is the key element in Skinner's S-R theory. A reinforcer is anything that strengthens the desired response. It could be verbal praise, a good grade or a feeling of increased accomplishment or satisfaction. The theory also covers negative rein forcers -- any stimulus that results in the increased frequency of a response when it is withdrawn (different from aversive stimuli -- punishment -- which result in reduced responses). Educationists and learning theories have given a great deal of attention was given to schedules of reinforcement (e.g. interval versus ratio) and their effects on establishing and maintaining behaviour. The reinforcement theory was applied to classroom instruction. Markle, 1969 and Skinner, 1968 outlines four steps in the application of this theory. These steps are: 1. Practice should take the form of question (stimulus) - answer (response) frames which expose the student to the subject in gradual steps. 2. Require that the learner make a response for every frame and receive immediate feedback . 3. Try to arrange the difficulty of the questions so the response is always correct and hence a positive reinforcement. 4. Ensure that good performance in the lesson is paired with secondary reinforcers such as verbal praise, prizes and good grades. The steps outlined above are the basis of the operant conditioning. According to Cherry (2005), “Operant conditioning (sometimes referred to as instrumental conditioning) is a method of learning that occurs through rewards and punishments for behaviour.” There are four types of operant conditioning namely positive reinforcement, negative reinforcement, punishment and extinction. According to Skinner’s terminology, any form of incentives such as goals and rewards may be referred to as positive rein forcers and the receiving of the reward or achieving the goal is termed as positive reinforcement (Skinner B.F., 1969). In positive reinforcement, a particular behaviour is strengthened by the consequence of experiencing a positive condition. Positive reinforcement is effective and largely used for two separate reasons. Firstly, it is one of the most powerful techniques available for the direction or motivation of the actions of other people. The second reason which is more philosophical is the versatility of the concept of reinforcement as an explanation of behaviour (Walker. S, 1975). In other words, the question is why do people behave the way they do? The answer will be that it is because they (people) are reinforced for it (Walker. S, 1975). In a similar fashion, a negative reinforcer is a stimulus one would desire to avoid. The act of escaping or avoiding a negative reinforcer is termed as negative reinforcement. Disincentives are punishers. There is a natural tendency to put punishments under the wing of negative reinforcement. However under the definitions of operant conditioning, negative reinforcement is the strengthening of a particular behaviour by the consequence of experiencing a negative condition Helping learners to become Meta-cognition in Mathematics Unlike other subjects, mathematics has a very wide base of practical application in our daily activities and everyday activities. This phenomenon actually makes mathematics a living subject because it is part of our everyday living. The teaching and learning of mathematics therefore involves relating “one’s knowledge to one’s own cognitive processes” (Flavell, 1976, p. 232) in other to have a firm understanding of concepts and apply them in the everyday routines of life. Metacognition, or “the seventh sense” (Nisbet and Shucksmith, 1984) has only relatively recently become a focus of research as an influential variable in the learning process and as a solution to enhancing the practicality of mathematical learning. The term “metacognition” is subject to differing definitions, interpretations and levels of understanding. Garner (1987:21) adapting Flavell’s 1981 “fuzzy concept” model, identifies the components of metacognition as cognitive goals, metacognitive knowledge, metacognitive experiences and strategy use. One of the issues within this “fuzziness” is deciding upon the boundaries between cognition and metacognition. Garner’s definition is that “metacognition is essentially cognition about cognition.” He follows Flavell’s differentiation between metacognitive knowledge and metacognitive experience, the former referring to knowing about metacognition and the latter referring to knowing how and when to use it and evaluating one’s choices, but warns that it risks becoming “an abused concept” (Garner, 1987:15 -16). McGilly (1994:279-280) states simply that “Metacognition is the ability to think about thinking, to be consciously aware of oneself as a thinker, and to monitor and control one’s mental processing. It is our awareness of ourselves (and by extension, others) as intelligent thinking creatures.” The relation of these explanations of metacognition to the teaching of mathematics at the primary is that, students should be given an opportunity to start developing understanding for mathematical concepts from a conscious reasoning perspective where they see themselves as the custodians of the thinking process. Primary school students should not be made to recite or memorize mathematical ideas without critical thinking. This is because when the conscious awareness of oneself as a thinker is absent, the mathematics class only becomes an environment where information is passed on from the teacher to the student and not a place where learning actually takes place. In anticipation to promote the effective use of metacognition in the mathematics class, several researchers have propounded learning styles, assessment orders, teaching theories and so on to aid mathematics teachers at the primary level to discharge their duties effectively. Sternberg (1990) links “learning styles” with metacognition in an examination of “mental self-government”. Cross and Steadman (1996:60-61) list a series of “cognitive” skills such as “rehearsal” and “comprehension monitoring” and distinguish them from metacognitive skills by stating that “Well-developed metacognitive skills enable students to determine how and when to apply cognitive learning strategies”. When given a task in school, however, children are very likely to jump into the problem with one strategy, continue the strategy without "looking back," and finish without re-examining the solution. Often, the result can be a misunderstood problem, or an ineffective strategy, and/or a solution that does not work. Meta-cognition emerged as an important mental activity for solving problems when researchers began to study children's intelligence and problem solving. Early analyses of problem-solving performance revealed that good or expert problem solvers tended to plan, monitor, and evaluate their thinking during problem solving more often and more efficiently than did poor or novice problem solvers (Flavell,1976). Meta-cognition is also important because it can increase the meaningfulness of students' classroom learning. (Schoenfeld, 1987). It can help students think of mathematics as a part of their everyday lives, help them make connections between mathematical concepts in different areas, and help them build a sense of a community of learners working together. Schoenfeld asserts that creating a "mathematics culture" in a classroom is the best way to develop meta cognition. Such a culture involves solving unfamiliar problems with your students, putting them on the board and working on them together. To develop metacognition through classroom discussions it is not enough to simply ask students to share their strategies and explain how they did it. Teachers need to have a clear mathematical agenda for each discussion, which often means choosing beforehand a certain strategy or two that they want to get discussed. By asking these questions in discussions, teachers are helping students to see what is valuable about a strategy besides its accuracy: efficiency, a clear written explanation, flexibility, a demonstration of conceptual understanding, etc. In the description of Dr. Edward de Bono’s six thinking hats and numeracy, he explains that A class brainsortm may uncover several reasons to choose particular methods that individual students may not have arrived at on their own. Once each option has been assessed for benefits and difficulties. Pohl’s suggested sequence for making choice is yellow hat, black hat, and red hat. Pohl further suggests a design sequence of blue hat, green hat and red hat for children exploring and inventing. Teachers are encouraged not shy away from comparing different students' strategies to make their points, though the classroom culture must support this first. When teachers and parents try to help students, it is important not to do too much thinking for them. By doing their thinking for the children they wish to help, adults or knowledgeable peers may make them experts at seeking help, rather than expert thinkers. On the other hand, by setting tasks at an appropriate level and prompting children to think about what they are doing as they successfully complete these tasks, adults can help children become independent and successful thinkers (Biemiller & Meichenbaum, 1992). In other words, it is often better to say, What should you do next?" and then to prompt the children as necessary, instead of simply telling them what to do. A teacher can also use KWL(K- What I know, W- What I want to learn, L – What I have Learnt) skills to facilitate Metacognition skills. Mevarech and Kramarski (1997) designed an instructional method, called IMPROVE, based on meta-cognitive guidance. IMPROVE is the acronym of all the teaching steps that constitute the method: Introducing the new material, Meta-cognitive questioning, Practicing, Reviewing, Obtaining mastery on higher and lower cognitive processes, Verification, and Enrichment and remedial. The meta-cognitive questioning includes four kinds of self-addressed questions: Comprehension question (e.g., What is the problem all about?), Connection question (e.g., How does the problem at hand similar to, or different from, problems you solved in the past? Please explain why), Strategic (e.g., What kinds of strategies are appropriate for solving the problem, and why?), and finally, Reflection (e.g., Does the numeric solution make sense? Can the problem be solved in a different way?). Each of these questions is based on rich literature in the area of meta-cognition (e.g., Schoenfeld, 1987; Lester, Garofalo, & Kroll, 1989; Pressley, 1986), and each has a unique contribution to the development of meta-cognitive processes and mathematical achievement (Mevarech, 1999). Metacognition can be facilitated by the use of inner speech, a kind of self-talk that enables students to direct and monitor their cognitive processing, and derive a deeper understanding and appreciation of their own thinking processes (Moffett, 1985). It is well known that young children often speak out loud while engaged in demanding activity (Flavell et al., 1997; Berk & Landau,1993; Englert et al., 1991); less understood is that older children and adolescents, and even adults engage in inner speech for similar purposes (John-Steiner, 1992; Tharp & Gallimore, 1988;Vygotsky, 1978). Yet, teachers frequently view self-directed speech as annoying, Distracting classroom behavior. Even when children do not self-talk out loud, they may be seen as inattentive, lost in their own world and absorbed by their own thought processes. In the view of Diehl, 2005; Flavell et al., 1997; Berk & Landau, 1993; Englert et al., 1991; and Rohrkemper, 1986, if teachers could envision self-talk as active constructivist activity intrinsic to metacognitive understanding, they could use inner speech as a tool to help students control and enhance their own cognitive performance. Hence, inner speech, this no cost underutilized educational resource, can be readily incorporated by teachers into their classroom instruction to improve student performance in many areas. Problem-solving is, in the first instance, a way of thinking, of analyzing situations, of using skills to reason out what cannot be learnt bymemorizing specific facts, but by absorbing oneself in the problem-solving process and applying existing experiences and existing knowledge to the problem that has to be solved (Schoenfeld, 1985a; 1992). Learning facilitation in mathematics is regarded as 'problem-solving' in which metacognition plays a well-defined role since problem-solvers, by default, become involved in cog-nitive and metacognitive behavior when they attempt to solve problems. Problem s are solved in three stages, namely , planning to solve the problem ;the actual solving of the problem; control, evaluation of, and reflection on, thesolution (Artzt & Arm our-Thomas, 1992). My experience: The subject ‘Mathematics’ normally bring shivers down the spine. Why? Let me start to reasoning out. Firstly, numbers are to very confusing, operations are too much complicated, lots of mental calculation to be done at a faster pace and above all the way it has been taught by the teacher in the classroom to the learners. I was lucky enough, as my tutor could help me learn mathematics in the way I understood. In other terms I can put it, teaching according to my learning style. This approach made me like mathematics so much that it gradually became my favorite subject. That is the main reason why I took up to teaching mathematics to the IB PYP learners and I actually made sure to undertake appropriate planning, development and reflection processes that support the achievement of PYP/MYP/DP learning outcomes. Mathematics, in my view is not just about calculations or operations but it acts as a tool of scrutinizing each and every problem and then choosing the right answer from the many multiples. It basically helps one to think and exercise their brain to an extent and which indeed helps us to make a correct decision in our life. This view is strongly supported by the National Council of Teachers of Mathematics, NCTM (1999), who posits that “through problem solving, children learn that there are many different ways to solve a problem and that more than one answer is possible. It involves the ability to explore, think through an issue, and reason logically to solve routine as well as non-routine problems.” Howard Gardner has identified Logical/mathematical as one of the eight (or more) intelligences that people have. As with the other intelligences in Gardner's classification system, people vary considerably in the innate levels of mathematical intelligence that they are born with. I normally use the thinking-aloud protocol while teaching mathematics and even do instruct the children to do the same. This thinking-aloud prompts children to think aloud during the whole problem solving. At the level of the students who are in the primary school, verbalization methods are the only ones that allow a glimpse into the participant’s mind during learning. This is to say that the teacher becomes away of the students, metacognitive ideas when the student voices this out Another form by which I promote metacognitive learning is to promote self-monitoring or leaner-assessment strategies. One form of self-monitoring involves students quizzing themselves in order to gauge comprehension; awareness of a lack of comprehension may prompt the student to try relearning the information. Additionally, self-regulation entails knowing when and how hard to work in order to acquire information (Zimmerman 1998). Self-regulators can decipher when it is prudent to spend additional time and effort to accomplish a task, as well as knowing when such additional work would produce little benefit. Finally, self-regulation includes having an understanding of whom to seek out for help (e.g., Newman 1994). This involves asking others for help when the student is aware that he or she does not have sufficient comprehension and another person would help them acquire a better understanding. When the thinking aloud method is employed, participants are asked to talk aloud during thinking, problem solving, and/or learning, and these verbal protocols are analyzed by means of coding schemes (Afflerbach 2000; Chi 1997; Ericsson and Simon 1993; Pressley and Afflerbach 1995; van Someren et al. 1994). The aim is to identify cognitive and metacognitive processes underlying task performance in different subject areas and contexts, but I will only concentrate on math only. One day I had given the children mixed sums with operations like addition, subtraction, division and multiplication. When they began to solve it they started to think aloud. While thinking aloud for solving addition problem, instead of adding he did multiplication and in a few minutes got the answer and wrote it down. On asking him the reason for multiplication he answered that multiplication is nothing but repeated addition, by doing this he could solve it much faster than the addition if he would have done. This showed that he understood the connection between addition and multiplication On another occasion I had instructed the class to solve any page from their math textbook according to their preference or likings and one child was struggling with the problem. The problem stated: if you multiply two numbers, you get the number sixteen. If you add those two numbers you get the sum of the number ten. What are the two numbers? On thinking aloud he tried saying 8 times 8 =16 but 8 + 8 = 16 and not 10. Then he tried thinking to get the answer 10 by thinking aloud that 5 +5 = 10 but the other condition deriving the number 16 is not solved. He tried all the math operation to solve the problem and then suddenly it was like the similar situation of that of Archimedes discovering and exclaiming ‘Eureka Eureka’ for him. He excitedly told me “Miss I got it, the two numbers are 8 and 2. “He told me that 8 times 2 =16 so the first condition is solved. Then 8+2= 10, so the second condition is also solved. This way, there was clear evidence of metacognitive skills. As a matter of fact, this fate was achieved through critical reflection of my lesson preparation as well as previous teaching and assessment activities used. I also reflected on my teaching and learning resources to ensure that they fitted the needs of learners. The demand to cover central ideas concerning mathematics education in a challenging and thorough way, and to stimulate pupils’ thinking processes about mathematical matters, has been present in math didactics literature throughout the last two decades. Thereby, the term “reflection” is frequently used. Kilpatrick (1986, p. 8) describes how the connotation of this term, originally used to depict physical and geometric phenomena, changed and now serves as a meta-phor for a variety of cognitive processes. Sjuts (1999a, p. 40) specifies “reflection” as “comparing and scrutinizing cogitation, thinking, and examination, directed to the matter at hand, which is characterized through differentiation, detachment, and deepening.” One can find other descriptions like “to engage in soul-searching”, “to pass in revue”, as well as “to re-late things”. Thus, “reflection” is used to describe a particular kind of high-level cognitive thinking process. Basically metacognition helps learners to go through its three elements and thus making the learners life-long learners. According to the North Central Regional Educational Laboratory, Metacognition consists of three basic elements: Developing a plan of action Maintaining/monitoring the plan Evaluating the plan Before - When you are developing the plan of action, ask yourself: What in my prior knowledge will help me with this particular task? In what direction do I want my thinking to take me? What should I do first? Why am I reading this selection? How much time do I have to complete the task? During - When you are maintaining/monitoring the plan of action, ask yourself: How am I doing? Am I on the right track? How should I proceed? What information is important to remember? Should I move in a different direction? Should I adjust the pace depending on the difficulty? What do I need to do if I do not understand? After - When you are evaluating the plan of action ask yourself: How well did I do? Did my particular course of thinking produce more or less than I had expected? What could I have done differently? How might I apply this line of thinking to other problems? Credit - Excerpted from Strategic Teaching and Reading Project Guidebook. (1995, NCREL, rev. ed.). - My research exposed me to the term group metacognition and I have since integrated the term to my teaching. I got a total glimpse of Group metacognition, as an essential element of mathematical problem solving within a group. Effective group mathematical problem solving involves not only finding a solution but also Metacognitively monitoring the group’s problem solving activity (Goos, Galbraith,& Renshaw, 2002). However, most research on metacognition has looked at the role of metacognition as an individual learning process (Flavell, 1976; Pugalee, 2001; Schoenfeld,1987; Schraw, 2001). The corpus of knowledge about group problem solving and learning indicates that students’ learning in successful groups can achieve higher cognitive levels than working alone (Johnson & Johnson, 1999). Group members can share ideas, develop common goals,as well as learn from and support each other’s learning (Benjamin, Bessant, & Watts,1997). Solving a mathematical problem as a group gives students access to a wide range of thinking strategies, contributes to students’ understanding of the problem, and provides alternative solutions (Cohen, 1994; Gillies, 2000). Group learning improves students’ mathematical understanding as well as improving their communication and group skills (Haller, Gallagher, Weldon, & Felder, 2000). During my very first session of encouraging, rather incorporating group strategies in mathematics classes , first I was a bit nervous about the session thought that the class would be in chaos as there will be children shouting, talking and making noise and it would be hard to control their discipline. But when I observed peer tutoring, brainstorming, monitoring each behaviour , thinking aloud, metacognition and tolerance and being sensitive towards individual’s needs , taking its immediate place in the group discussion , made my lesson go through very smoothly and at the end each and every child understood the lesson well, here, my learners and I witnessed chaos but “Learning chaos” which was appreciated and embraced with open-mindedness , a quality of the Learner Profile. To this end, my observation raised further questions on metacognition especially as far as classroom management and control are concerned. This is an important point for a teacher to observe as the absence of a very good classroom management plan review can turn the classroom into an atmosphere of uncontrolled chaos in the bid to promote metacognition. Conclusion Metacognition is a term that deserves a higher profile in the classroom and in professional development. The Importance of knowing what you know, unless you are taking a test or playing Jeopardy, metacognition is more important to success than cognition. In real life, when you’re faced with a question the first decision is whether you know the answer or not. With strong metacognitive ability this is easy. If you know the answer, but can’t come up with it, you can always do a bit of research. If you know for sure that you don’t know, then you can start educating yourself. Because you’re aware of your ignorance, you don’t act with foolish confidence. The person who thinks they know something that they really don’t makes the worst decisions. A person with poor cognitive ability, but great metacognitive ability is actually in great shape. They might do poorly in school, but when faced with a challenge they understand their abilities and take the best course of action. These people might not seem intelligent at first glance, but because they know what they know, they make better decisions and learn the most important things. Many influential documents in mathematics education, such as the NRC's report "Adding it Up", discuss the idea of metacognition without actually naming the term. The NCTM's Principles and Standards does mention metacognition in discussing problem-solving, citing research that indicates "students' problem-solving failures are often due not to a lack of mathematical knowledge but to the ineffective use of what they do know" (p.53). Teachers need to recognize the importance of metacognition and make developing reflective habits an explicit goal in their classrooms. References: Chalmers C 2008, Group Metacognition During Mathematical Problem Solving, Queensland University of Technology, accessed 6 July 2010 Cherry K 2011, Introduction to Operant Conditioning. Accessed 7 March 2011 Cherry K 2011, What is Cognition?, Accessed 4 March 2011 Fox W 2009, What is Relational Learning?, Relational Institute, accessed 6 March 2011 Investigations 2009, Getting Metacognition Out of the Close, accessed 9 May 2010 Jones M.H, Estell D.B & Alexander J.M 2007, Friends, classmates, and self-regulated learning: discussions with peers inside and outside the classroom. E-Journal, accessed 4 Junes 2010 Junior Achievement Michiana 2007, Elementary Student Achievement, accessed 7 March 2011 Kaune C 2006, Reflection and Metacognition in Mathematics Education, accessed 31 May 2010 Mevarech Z 2006, EMPIRICAL STUDY. The effects of IMPROVE on mathematical knowledge, mathematical reasoning and meta-cognition, accessed 9 June 2010 Nate K 2009, Metacognition in Humans and Animals, Current Directions in Psychological Science, Blackwell 18:1; 11-15. National Council of Teachers of Mathematics (NCTM), 1999, What is Mathematics, Archived Information, accessed 3 March 2011 Nicholls H 2005, Cultivating “The Seventh Sense” – metacognitive strategising in a New. Zealand secondary classroom, accessed 7 June 2010 Pohlman C 2009, Knowing Thyself: The Importance of Metacognition, accessed 9 May 2010 Skinner B.F 1950, Operant Conditioning, accessed 5 March 2011 The North Central Regional Educational Laboratory, Metacogtition, accessed 7 March 2011 Trent 2007, How Much Cash Is Appropriate To Carry? accessed 8 June 2010 Essential Questions related for the essay Metacognition 1. How have you demonstrated your ability to undertake appropriate planning, development and reflection processes that support the achievement of PYP/MYP/DP learning outcomes? – page 11 2. How have you demonstrated your ability to design learning activities and teaching strategies that support the achievement of PYP/MYP/DP learning outcomes? – page 12 and 13 3. How have you demonstrated your ability to differentiate teaching and learning activities to meet the needs of all students? – page 13 The essential questions that relate to the Professional learning area of enquiry in the Level 1 Teacher award unit are as follows Essential questions that related to the Professional learning area of enquiry 1. How have you demonstrated your ability to undertake critical evaluation of planning, teaching and assessment activities, and learning resources to assess their impact on student learning and where necessary revise PYP/MYP/DP practices? – page 13 2. How have you demonstrated your ability to participate in and undertake critical evaluations of collaborative working practices intended to promote PYP/MYP/DP learning outcomes? – page 12 3. How have you demonstrated your ability to participate in and evaluate the use of information and communication technologies to engage within the wider IB community to further develop PYP/MYP/DP learning outcomes? – page 6 Read More