AUSTRALIAN ASSOCIATION FOR RESEARCH IN EDUCATION
1998 ANNUAL CONFERENCE
ADELAIDE
Metacognition in Mathematical Problem Solving
Ban-Har Yeap
Raffles Institution, Singapore
banhar@pacific.net.sg
Abstract
The main purpose of the study is to understand the role of metacognition in mathematical problem solving to generate a set of recommendations for classroom instruction. The three specific aims are, firstly, to observe patterns of behaviours when students solve mathematical problems, secondly, to observe the types of metacognitive behaviours during mathematical problem solving, and thirdly, to understand the role of metacognition during problem solving. The thinking aloud method was used to generate data from a group of Year 7 students. Data analysis using a previously developed taxonomy of problem solving behaviours and Flavell's model of cognitive monitoring provide insights into metacognitive aspects of mathematical problem solving. This paper outlines the main research findings.
Keywords
mathematical problem solving, metacognition
Introduction
Since the 1980s mathematics curriculum in many countries have emphasized the central importance of problem solving. Only in the 1990s mathematics curriculum in Singapore schools was revised to make the development of problem solving ability its primary aim.
A problem is a situation confronting an individual that requires a resolution, a situation that he wants or needs to achieve resolution, one whereby he has enough resources to initiate and continue a path towards resolution which is not immediately obvious. To complete this path he needs to use and manage his resources well. Such an act of resolution is problem solving. Metacognition has been identified as a key factor in the problem solving process. Metacognitive knowledge is that part of one's knowledge that has to do with people as cognitive agents and with their cognitive tasks, goals, actions and experiences. Metacognitive experiences are conscious experiences related to any aspect of a cognitive enterprise. Metacognition is a collective term used to include metacognitive knowledge, metacognitive experiences and the interaction between the two.
This paper outlines a research study that aims to understand the role of metacognition in mathematical problem solving and its main findings, which will be used to propose a set of recommendations for mathematics instruction to promote metacognition and problem solving.
The Present Study
The main purpose of the study is to investigate the role of metacognition in mathematical problem solving. The specific aims are, firstly, to observe patterns of behaviours when students solve mathematical problems, secondly, to observe the types of metacognitive behaviours during mathematical problem solving, and thirdly, to understand the role of metacognition during problem solving.
Subjects
The subjects are ten students selected from an intact class of forty Year 7 students in a secondary school in Singapore. The forty students took a problem solving test shown in Figure 1. Each response was classified as successful, unsuccessful or partially successful. A successful response is one where a complete solution set was obtained. An unsuccessful response is one where either no response was recorded or where the solution set was incorrect and the method used unsuitable. A partially successful response is one where a suitable method was used but the solution set was either incorrect or incomplete. The purposive sample comprises students who were successful as well as those who were not in the problem solving test. Among the ten subjects, four were successful in all three problems, two were unsuccessful in all three problems while the other four were successful in some problems and not in others. The ten subjects were then trained to use the thinking aloud method to solve problems.
Data Collection
Each of the ten subjects was asked to solve the problem in Figure 2 using the thinking aloud method. Each problem was presented to the student on a piece of paper. The instruction to verbalize anything that comes to his mind and not to explain his behaviour was explicitly given. The researcher observed the subject as he solved the problem by thinking aloud. The verbal responses were audio recorded. The audio recording of each subject's verbal response was transcribed to produce a verbal protocol. After the subject had solved the problem, the researcher interviewed him to obtain an explanation to the method used to solve the problem as well as to confirm some observations made by the researcher during the problem solving process. The data collected included (1) verbal protocols, (2) written response, (3) field notes from researcher's observation, (4) field notes from interview sessions. The verbal protocol, together with the other three data sources, were used to construct a more complete problem solving protocol for each of the ten subjects. Each problem solving protocol was segmented into episodes. Each episode comprises one consistent type of behaviour.
INSERT FIGURE 1 ABOUT HERE [HARD COPY PROVIDED]
STARS
What numbers can replace the stars? Each star represents one digit.
x *
_____________
4 *
4_____________
Figure 2 The problem used during data collection
Data Analysis
The problem solving protocols were initially analyzed using a taxonomy modified from the one previously derived by Foong (1993) in her study on a group of Singapore trainee teachers solving problems similar to the one used in the present study. There are five categories of problem solving behaviours in the modified taxonomy. Table 1 describes the modified taxonomy.
Table 1 Taxonomy of problem solving behaviours based on Foong's taxonomy (1993)
Types of behaviour Code
Problem orientation behaviours (P-category) P
Initial reading of the problem
Re-reading of the problem
Paraphrasing while reading the problem
Examining conditions of the problem
Constructing representation of the problem
Heuristic problem solving behaviours (H-category) H
Stating an answer
Looking at a particular case in a random manner
Looking at a particular case in a systematic manner
Making a generalization
Making a deduction
Checking computation
Recalling similar problem
Drawing diagrams
Making a guess and check
Searching for pattern
Domain-specific behaviour (K-category) K
Performing computation/procedure
Recalling a fact
Types of behaviour Code
Affective behaviour (A-category) A
Demonstrating negative self-evaluation
Giving up
Demonstrating emotional expression
Metacognitive behaviours (M-category)
Stating a plan M1
Clarifying task requirements M2
Reviewing progress M3
Recognizing error M4
Detecting new developmet M5
The decision to use this taxonomy was largely due to the similarities in linguistic background of the sample and the types of problems used in the present study and those in Foong's study. The data analysis using this taxonomy generated a catalogue of observable behaviours that can be recognized as metacognitive in nature.
Figure 3 shows how a part of one subject's protocol was coded using the modified Foong's taxonomy.
Protocol B9 Xuexin solving STARS.
Verbatim Transcript Code
[reads the problem statement] P
what times what equals 4 P
2 times 2 .. 4 equals 2 times 2 P
or M5
find last digit equals 4 M1
6 times 4 H
4 equals to 1 times 4 M3 ah .. last digit plus second digit P
so the first number could be sixty H
4 times 6 again should be .. 24, 6 times 4 .. 24 M3
6 times 6 .. 36 .. 38 .. [384] K not correct M4
say, 404 .. H
404 divided by sixty something ah so should be about P
not 2, 3, 4, 5, 6, 7 H
could be 7 .. 42 star H
so maybe it's 424 H
times digit is 6, 6 times something up to 424 P
sixty something times what gives 424 M3
Figure 3 Coding of a problem solving protocol using the modified taxonomy
The researcher considered several other ways to analyze the problem solving protocols. This was done for two reasons. Firstly, the researcher wanted to examine the data from a different perspective. Secondly, the researcher wanted further insights into the role of metacognition during problem solving. After several months of deliberation and constant discussion with other researchers, the model of cognitive monitoring (Flavell, 1981) was eventually adopted to provide another perspective into the role of metacognition during mathematical problem solving.
The model of cognitive monitoring comprises four interacting components (Figure 4). The four components are cognitive goals, cognitive actions, metacognitive experiences and metacognitive knowledge. The arrows show that each component can influence all other components.
INSERT FIGURE 4 ABOUT HERE [HARD COPY PROVIDED]
Cognitive goals (CG) are explicit objectives that instigate and sustain a cognitive enterprise. Cognitive actions (CA) or strategies are those taken to achieve cognitive goals of a cognitive enterprise. Metacognitive knowledge (MK) is knowledge about people as agents in a cognitive enterprise as well as knowledge about tasks, goals, actions and experiences during a cognitive enterprise. Metacognitive experiences (ME) are experiences (ideas, thoughts, feelings or sensations) related to any aspect of a cognitive enterprise. The cognitive enterprise was to determine possible numbers that could replace the stars in the given mathematical problem (Figure 2).
Figure 5 shows how a part of one subject's protocol was coded as a component of the model of cognitive monitoring. The coded protocols are then used to generate cognitive-metacognitive maps (Yeap, 1997) to show how each component influences the others. The maps essentially describe prominent types of influence for each subject.
Protocol C9 Xuexin solving STARS.
Verbatim Transcript Code
[reads the problem statement] CA
what times what equals 4 CG
2 times 2 .. 4 equals 2 times 2 CA
[that the goal is incomplete] ME
or find last digit equals 4 CG
6 times 4 CA
[that his initial action was incomplete] ME
[reviewed his initial action] 4 equals to 1 times 4 CA [that he has rectified the incomplete action and could proceed] ME ah .. last digit plus second digit CG
so the first number could be [64 times 6] CA
4 times 6 again should be .. 24
6 times 4 .. 24
6 times 6 .. 36 .. 38 .. [384]
not correct ME
[that alternative strategy should be explored if existing one fails] MK
should try [that an alternative strategy should be considered] ME
say, 404 .. 404 divided by sixty something ah so should be about CG
[that a guess and check method could be used] MK
not 2, 3, 4, 5, 6, 7 CA
could be 7 .. 42 star
so maybe it's 424
times digit is 6, 6 times something up to 424 CG
sixty something times what gives 424
Figure 5 Coding of a problem solving protocol as components of Flavell's model of cognitive monitoring
Results & Discussion: A Catalogue of Metacognitive Behaviours
Metacognitive behaviours detected from the problem solving protocols are of five categories. The behaviours are (1) stating a plan, (2) clarifying task requirements, (3) reviewing progress, (4) recognizing error, and (5) detecting new development.
M1-type behaviour involves stating a plan of action. One subject stated his plan to find digits that when multiplied give a 4 or a number ending with 4.
ah so the both stars multiplied must get about 4 P
the last digit must get 4 P
find out.. find out all the possible numbers now M1
M2-type behaviours involves clarification of task requirements. One subject checked if the stars represented the same or different digits. He is also concerned if more than one answer was required.
but the stars right? They must be the same number? M2
so ah.. you want how many numbers? M2
M3-type behaviour involves reviewing progress. Some review episodes were followed by a consequent actions while others were not. One subject reviewed his work, and observed and corrected an error.
about 2 or 4 H
and the second one will most probably .. the first one will be 1 .. M3
will be either 1, 2 or 4 H
M4-type behaviour involves detecting errors in reading, in comprehension, in computation, in executing of a plan, and in planning. One subject realized that the factors of 14, 24, 34, etc. that he was finding should be single digit numbers.
ah 14 equals to 2 times 7
14 equals to 1 times 14 H
but can't .. one digit M4
so 24 equals 6 times 4 .. 24 equals H
M5-type behaviour involves detecting and using new development to get closer to the solution. One subject, after attempting to use factors of 4 unsuccessfully to find a solution, stumbled upon the fact that he could also consider the factors of numbers ending with 4.
66 .. 66 times 4 H
is 4 .. 2 .. 6 times 4 .. 24 .. 26 .. 26 K
the number behind might be 4 M5
so 6 .. 6 .. 67 times 2 H
The main purpose of documenting the existence and types of metacognitive behaviours when students were solving mathematics problems is to derive a catalogue of verbal utterances which aims to serve as an empirical framework for classroom teachers to detect the occurrences, or absence, of metacognitive processes amongst students. Such a catalogue has three immediate classroom applications. Firstly, teachers could help students capitalize on metacognition to be effective problem solvers. The teacher is also able to detect an effective use of metacognition and highlights such usage to students. Secondly, teachers could discriminate classroom activities and tasks which promote, or are not conducive for, the development of metacognition. Armed with a knowledge of metacognition in oral form, teachers complemented with a wealth of classroom experiences are able to reflect upon whether an activity would elicit metacognitive-type responses. Thirdly, teachers could use the catalogue to derive a framework for assessing metacognitive aspects of problem solving. A checklist could perhaps be derived based on the derived catalogue of metacognitive behaviours for use during classroom interactions or during oral problem solving sessions.
Results & Discussion: Cognitive-Metacognitive Maps
The generation of cognitive-metacognitive maps produces maps for each of the ten subjects which were then used to suggest the role of metacognition in mathematical problem solving. An example of each of the three categories of maps obtained in the present study is included in the Appendix.
Type X (Xuexin) maps suggest that the subject tends to set a cognitive goal which would influence his subsequent cognitive action. The cognitive goals are often changed or modified as a result of a cognitive action. Metacognitive experiences are often sparked off by his cognitive actions. Type Y (Chee Seng) maps suggest that the subject tends to have metacognitive experiences as a result of performing a cognitive action. Such experiences as well as metacognitive knowledge in turn influenced subsequent cognitive actions. Type Z (Adrian) maps suggests that the subject tends to formulate goals based on a cognitive action or a metacognitive experience, tends to have metacognitive experiences as a result of an action, and performing an action as a result of setting a cognitive goal or having a metacognitive experience.
The similarity among the three types of cognitive-metacognitive maps is that all of them show that metacognitive experiences are often sparked off by cognitive actions. However, in Type X maps the metacognitive experiences do not influence subsequent stages in the problem solving process significantly. In Type Y maps, metacognitive experiences dictate subsequent cognitive actions significantly. In Type Z maps, metacognitive experiences not only dictate cognitive actions significantly, they are also of significant influence to the formulation of subsequent cognitive goals. Metacognitive knowledge is conspicuously not significant in the problem solving process of all the ten subjects in the present study.
Three roles of metacognition emerged from scrutinizing the cognitive-metacognitive maps of the ten students. Firstly, the generation of metacognitive experiences seems important. When one has metacognitive experiences and knows how to capitalize on these experiences, there is a higher chance that a problem solving endeavour will be successful. The occurrence of metacognitive experiences on it own does little to ensure successful problem solving. The generation of a metacognitive experience is often sparked off by a cognitive action. It is more rare that the setting of cognitive goals would lead to metacognitive experiences. It is more common that the execution of a cognitive action after the setting of a cognitive goal prompts a metacognitive experience.
Secondly, the possession and retrieval of metacognitive knowledge when they needed are important determinants of problem solving attempts. The retrieval of such knowledge for identifying a cognitive goal and performing a cognitive action often helps to facilitate a more efficient problem solving attempt. It is interesting to observe that the little metacognitive knowledge demonstrated by the subjects in this study was a reflection of the classroom interactions they have experienced.
Thirdly, the influence of metacognitive experiences and knowledge seem significantly important to determine a successful problem solving attempt. Cognitive goals set guided by metacognitive experiences and metacognitive knowledge tend to play an important role in determining success in problem solving. It is also common to observe that cognitive actions guided by metacognitive experiences and knowledge tend to produce good results in solving problems.
Conclusion
In summary, the findings confirm the importance of metacognition in mathematical problem solving. It is observed that metacognition provides a more promising platform to set goals, and to perform actions to achieve those goals, during problem solving. As such the development of metacognitive skills must not be overlooked during instructions.
It was also observed in the present study that subjects often demonstrated metcognitive behaviours. The use of modified Foong's (1993) taxonomy described the types of metacognitive behaviours commonly shown by students. Flavell's (1981) model, however, showed that most of the metacognition that subjects had were in the form of metacognitive experiences. Metacognitive knowledge was far less common in occurrence. However, whatever little metacognitive knowledge that was possessed and retrieved was seen to be important in mathematical problem solving. The mere generation of metacognitive experiences was, however, observed to be insufficient in ensuring success during problem solving. The processes following metacognitive experiences were just as important. Thus, instructions must help students gain and manage metacognitive knowledge, and provide them with opportunities to manage their metacognitive experiences.
References
Flavell, J. H. (1981). Cognitive monitoring. In W. Dickson (Ed.), Children's oral communication skills. New York: Academic Press.
Foong, P. Y. (1993). Development of a framework for analyzing mathematical problem solving behaviours. Singapore Journal of Education, 13(1), 61-75.
Yeap, B. H. (1997). Mathematical problem solving: A focus on metacognition. Unpublished M.Ed. dissertation. Nanyang Technological University, Singapore.
Appendix
Cognitive-metacognitive maps of three subjects
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