“The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.”
Cassius, in Julius Caesar (I, ii, 140-141)
: )
Metacognition and mathematical problem solving
Metacognition is a crucial element in problem-solving, which is itself a key component in mathematics learning. To monitor and regulate one’s cognitive processes in problem-solving, Polya (1945) describes a four-step method: first, one has to comprehend the problem by sub-dividing it into more manageable parts and recognize any given data, conditions and variables to be found; second, one devises or selects a strategy to find the connections between the known data and the unknowns to be found; third, one executes the plan, scanning, regulating and examining each step; and, finally, after obtaining the solution, one evaluates the results which may involve re-visiting the previously taken steps.
Building on Polya’s work, Schoenfeld (1987) describes effective mathematical problem-solving as being contingent on how one uses four types of knowledge/skills: (1) resource knowledge, which is knowledge about one’s abilities and cognitive processes including knowledge of how to perform tasks or procedures; (2) heuristics, which are specific problem-solving methods or strategies; (3) regulatory processes, which includes the organisation and selection of resources and strategies; and (4) beliefs, which includes perceptions of and assumptions about mathematics (Gama, 2004).
Note: The above paragraphs are adapted from my minor research paper, Examining Supports for Metacognition in Singaporean Lower Secondary Mathematics Textbooks, NIE, 2011. All rights reserved.
Related links:
(1) Metacognition – The secret to learning and problem-solving;
(2) Metacognition enhances learning;
(3) Is Metacognition part of the Singapore Math curriculum?
_______________________________________
TUITION CLASSES:
_______________________________________________________________
EDUCATIONAL SERVICES:
______________________________________________________________
By EX-MOE TEACHERS & EXPERIENCED TUTORS
@ BLK 644, BUKIT BATOK CENTRAL, #01-68. S(650644).
CALL 65694897 OR SMS 98530744 OR 97860411.
Metacognition – The ‘secret’ to learning and problem-solving …
You don’t have to be ‘good in English but bad in Math’, or ‘good in Science but lousy in History or Literature’, or ‘good in Chinese but lousy in English’. It doesn’t have to be a one-or-the-other kind of thing. Every subject has its own way of learning and thinking about, but you can only discover it if you bring yourself one level up – think about the thinking itself, and think about how you learn. You can only learn how to learn if you think about learning. You can only learn how to think if you think about thinking.
Thinking about thinking has another name: metacognition, which formed the basis of my minor research work completed in 2011 at the NIE. A lot of research into metacognition centers around mathematical problem-solving, although metacognition can be applied equally well to other subjects. So for the purpose of illustration here, I will use math as the basis for our discussion about metacognition.
What is Metacognition?
In its simplest form, metacognition can be described as thinking about thinking (Wellman, 1985). Flavell (1976) has stated some examples of what constitutes metacognition:
I am engaging in metacognition if I notice that I am having more trouble learning A than B; if it strikes me that I should double-check C before accepting it as a fact; (…)if I become aware that I am not sure what the experimenter really wants me to do; if I sense I had better make a note of D because I may forget I; if I think to ask someone about E to see if I have it right. (p. 232).
The above sounds familiar, doesn’t it? Yes, you may have been practising metacognition sub-consciously, all your life, without realising it has a formal name, or without realising how important it is to learning and thinking.
Research in metacognition has examined various forms of cognitive processes by building on Flavell’s initial notion of metacognition as the monitoring, regulation and arrangement of thinking processes to achieve specified goals (Gama, 2004).
For example, Brown (1987) argues that metacognition consists of two components: knowledge of cognition, and regulation of cognition. The former involves being aware of one’s cognitive abilities through self-reflection while the latter pertains to mental activities such as monitoring and controlling one’s thinking processes in the course of learning or problem-solving. According to Brown, although these two types of metacognition are distinct from one another, they are usually used together in the same cognitive process as one type often utilises the service of the other (Gama, 2004).
Note: The above paragraphs are adapted from my minor research paper, Examining Supports for Metacognition in Singaporean Lower Secondary Mathematics Textbooks, NIE, 2011. All rights reserved.
Related links:
(1) Metacognition and problem-solving;
(2) Metacognition enhances learning;
(3) Is Metacognition part of the Singapore Math curriculum?
TUITION CLASSES:
_______________________________________________________________
EDUCATIONAL SERVICES:
______________________________________________________________
By EX-MOE TEACHERS & EXPERIENCED TUTORS
@ BLK 644, BUKIT BATOK CENTRAL, #01-68. S(650644).
CALL 65694897 OR SMS 98530744 OR 97860411.
What makes my day …..
I’m doing relief teaching again today, at a school in Dover Road. I’m now in the library, waiting for the next class to begin at 2.25pm. Just now, after the Physics class ended, students for the next lesson started trickling in. One of them, a tall, lanky boy, looked at me and said,”Hey, weren’t you the teacher that took us for the Log class? That was 2010.”
Looking hard at him, I replied,”Yes, must be. And you look familiar too, except that I now remember you being shorter, but just as thin.” He smiled. “I really enjoyed that class.” “Me too”, I remarked before leaving.
Nothing pleases a teacher more than a meeting an ex-student who has benefitted from his or her teaching. Very often these students have grown up quite a bit and I don’t recognise some of them, but they would stop me in my tracks in a public place, or wave at me from afar and I would take some time to search my memory. Often I dont remember their names already, as so many batches have come and gone. Once a young man stopped talking to his girlfriend, stood up and said hi to me and we chatted for some 5 mins; I hope they are still together.
Sometimes I wonder where the rest have gone to, what they are doing now, and whether they are happy and successful, howsoever they define success. Some I found in facebook, and a few came back to help me tutor. Of the latter, three came in Sec 2, stayed until JC2, underwent NS, and now are undergrads, and I get to meet them every week and ask about their current lives. From all of them I learn about what matters to young people nowadays, their worries, their aspirations and their hopes for the future.
I hope parents spend more time with their children, for some of them tell me things they wouldn’t share with their parents. But I also often tell these youngsters to understand their parents, instead of always asking to be understood.
Parents have their own worries, fears and hopes too. I’ve advised parents before on how to handle their children, and vice versa, and when either party tells me things have become much better, that too, makes my day.
I also feel happy when I have induced a person to do self-reflection, because I do it everyday, in fact almost every moment, even when I’m teaching. That, I think, is the only way to improve. I mean, I don’t see how you can improve if you don’t know what you don’t know, and you also don’t know what you know. Thinking about your own thinking, often called metacognition, is very important in life. I’ll talk about this in another post.