Metacognitive Strategies in Solving Mathematical Problems - Article Example

This study has intended to investigate the role of metacognition with reference to problem solving in the field of calculus. The studywas developed and carried out on a model of metacognitive thinking strategies. This model was tested for its reliability in problem solving skills for non routine calculus problems. A questionnaire was administered among 480 undergraduate students who were randomly selected. The rate of return was 90%.By using principal component analysis the study has identified seven fundamental dimensions of metacognitive thinking strategies. These are 1) Self-efficacy, 2) Definition, 3) Exploration, 4) Accommodation, 5) Strategy, 6) Execution and 7) Verification. The research study has applied multiple regression analysis to evaluate the predictive ability of the identified variables so as to tabulate the performance for the routine and non-routine calculus problems. The study has significantly found that problem solving skill is acquired through practice and utilization of thinking strategies which is the corner stone on which advanced mathematical ideas, particularly calculus is built upon. The study has also revealed that there are six meaningful predictive factors for calculus related to performance in problem solving. The variable of Strategy is the major projection followed by Accommodation, Self-efficacy, Definition, Exploration and Execution. Nevertheless, the variables with the highest practical importance are Accommodation and Self-Efficacy. With these findings, educators will be able to clinically evaluate a person's ability to regulate, monitor and control his/her own cognitive processes. Instructional strategies can be developed for individuals having difficulty with the learning environment. The purpose of this study is to develop an instrument to adequately identify metacognitive strategies utilized by individuals' in the processes of solving mathematical problems. Introduction Definition and Meaning Cognitive Process: A cognitive process whether it is regular or irregular, conscious or unconscious is induced by means of metacognitive knowledge in a vicinity of active conscious memory. The thinking strategy for metacognition is essential for any extended activity, with special reference to problem solving. This is based on the hypothesis that a problem solver needs to be aware of the current activity regarding the overall goal along with the strategies used to attain that goal. It is well established that successful students possess powerful strategies for dealing with problems in order to arrive at novel solutions. Background Information on the Problem The metacognitive awareness merges itself by adjusting and recognizing the expansive bank of multiple metacognitive experiences. This has been previously described by Flavell (1976) as being a conscious cognitive or affective experience that accompanies our actions by dispensing to an intellectual enterprise. Thus, metacognition involves the "active monitoring and consequent regulation and orchestration" of cognitive course in order to achieve cognitive goals (Flavell 1976:p. 252). Review of Literature Kluwe (1982) has expanded on Flavell's theory of metacognition by shedding new light on the previously established empirical concept. He has identified two general attributes common to thinking procedures which are known as 'metacognitive'. Both attributes identified by Kluwe have to do with the person who exercises metacognitive thinking. The first postulates that "the thinking subject has some knowledge about his own thinking and that of other persons," while the second professes that "the thinking subject may monitor and regulate the course of his own thinking, i.e., may act as the causal agent of his own thinking" (1982, p. 202).Drawing from that, all processes seek to adapt and regulate a multitude of solutions actively. Research Question(s) Statement of the Problem Creation of Essential Question The concept of metacognition and thinking practices was first echoed by Rickard (1995) and his statement further added that exploring problems will build mathematical connections. To explore the problem successfully the students have to go through all the different permutation between the known and the unknown. This is demanding and it challenges the student's ability to reach deep down into his or her own knowledge reservoir and come up with all the possibilities. The sixth and final significant predictor was the ability to execute. This is the actual working out of the algorithms. A process-constrained task requires students to carry out a procedure or a set of routine procedures in solving the problem. Philipp, (1996) suggested that people tend to streamline the process by which we compute invent computational algorithms. However, what is important is that the student should in most cases work the question till the end and not resort to guessing or anticipating the solution. Hypothesis & Predictions The hypothesis for this research study is, "Students with higher levels of success rate in solving calculus problems tend to be frequent exploiters of metacongnitive thinking strategies." Methodology General Design of Study Regression analysis was run to estimate the predictor variables that play an effective role in metacognitive thinking strategies. A multiple regression analysis was run for six predictor variables (Self-Efficacy, Exploration, Definition, Accommodation, Strategy, and Execution) in Statistical Package for the Social Sciences (SPSS) program to determine the strength and predictive ability of the seven independent variables in explaining variations by students who use of metacognitive thinking strategies in their studies. Definition of Variables The study comprises of the following variables: Seven Predictor Variables: 1) Self-efficacy, 2) Definition, 3) Exploration, 4) Accommodation, 5) Strategy, 6) Execution and 7) Verification. Criterion Variables: Metacognitive Thinking Description of Study Sample The sample comprised of 232 (53.6%) male students and 201 (46.4%) female students. The second part of Table 2 summarizes the distribution of the respondents in terms of their academic achievement. The CGPA measure is used as an academic indicator. The figures obtained by the analysis suggest that majority of the students i.e. both males and females were reported to have CGPA score between 2.5 to 3.5. Fifty-six students were found to score less than 2.5, but more than 2.00. However, only a small number of participants (20) had a CGPA score below 2.00. Forty eight (48) students had a CGPA score of more than 3.50. The mean for the study sample was 3.03 with a standard deviation of 0.47. This indicates that the sample was significantly representative for all achievement levels. Note: The study was conducted by using the study sample of A-levels and first year undergraduate students therefore all the students are from the age group 18-20. Procedure A questionnaire was used to track the mathematical performance in students consisted of three calculus questions and 48 self-reported questions. The first batch of questions tested the basic differential level, whereas the second challenged the skills of each student in its successive application. The final batch of questions tested the student for his/her ability in problem solving. The preliminary test was administered to all students from the study sample. The result submissions were evaluated by two people simultaneously; a mathematician who has been handling calculus for more than 20 years whereas the results and performance was evaluated by the author(s). The scores were separately assigned by both examiners and were compared later on to tabulate a mutual performance table which is given in table 1. Table 1 Summary (Mean; SD) of the overall achievement scores of the respondents Evaluation Male (n = 232) Female (n = 201) Overall (n = 433) First Evaluation 39.45 (17.49) 35.659 (19.08) 37.69 (18.32) Second Evaluation 39.45 (17.43) 35.80 (19.13) 37.76 (18.31) The average marks scored by both male and female students were 39.45 and 35.65 respectively. Whereas the average mark was track for the whole slot was tracked to be 37.69. There was no significant difference between male and female students with respect to the average marks that they scored (Z =2.16; P< 0.01). The average scores given by the evaluators were also compared by using Z test so as to estimate the personal bias in the data. Results showed no statistical difference as being observed between the two evaluators with respect to the marks awarded (Z =0.00; P < 0.05). This preliminary analysis revealed that the evaluation carried out was unbiased and with minimal occurrence of errors. Examination and Study Accommodation "Accommodation" was found to be the most significant factor variable in a student's perception towards metacognitive thinking behavior. It accounts for 28.49% of the overall variance. This factor was based on the basis of the following observations. The first is the ability of the student to reorganize information within their schemata to meet specific requirements that are required to effectively attend to a question. This item is noted to be put into practice by an individual via the skill of exploration. The degree of exploration to further explain the results of the analysis are constructed on the grounds of three assumptions, i.e. (a) A student's ability to look back and review his/her past experience; (b) The respondent's ability to put together all the ideas the relative permutations; and (c) The aptitude and skill of a student to realize and come to terms with all the possible situations that can be derived from the question. When the questionnaire was initially constructed, the hypothesized 12 items were hypothesized to measure this factor. Nevertheless, in light of the rotated factor analysis, only nine items were retained. Definition Definition is the fourth factor which explains 5.13% of total variance. 11 items were designed for the successful measurement of this construction. The results were based on the rotated factor analysis and only four items turned out to be appropriate and retainable. The factor variable was developed on the basis of hypothetical criteria. The criterion deals with the general ability of the students to rephrase a given problem so that it matches with their own schemata. The second level of the theory incorporates the skill and capability of the students to reflect and match the domain specific vocabulary and definition for the prescription of algorithmic and mathematical models. Similarly, the third principle of the theory deals with the ability of the student to organize the available information meaningfully, whereas the final factor is a student's capacity to mentally represent the given situation in terms of graphical aids. Self Efficacy Self-Efficacy was found to be the fifth most significant factor that contributes in forming a student's perception towards metacognitive thinking behavior. This factor accounted for 4.97% of the total explained variance. Its occurrence can be explained on the basis of some commonplace academic behaviors. The first is that whenever the students find a particular subject, which in this case is mathematics to be enjoyable, they tend to score higher in that disciple. This would function because positive feelings towards mathematics will indirectly pave the way in enhancing the understanding and grasp of the subject matter. Correlation Analysis The analysis for the correlations revealed three of the six predictors to be statistically significant. Accommodation, Strategy and Self-Efficacy were statistically significant. In the model, students reported the use of thinking strategies and the main criterion variable was tested by using six predictor variables. Analysis of Variance (ANOVA) asserts the overall model to be statistically significant; F (6, 195) = 46.207, = 0.001, MSE = 146.3 71, and the set of the independent variables has accounted for 59% of the total variance explained. The adjusted coefficient of determination (adjusted R2 was .57, with an estimated standard error of 12.09. Further analysis of the predictive power of the individual predictors indicated that all the factors in the study were statistically significant. Reflection These seven components were able to explain around 60% of the total variance. Among these the higher variances were explained by Accommodate, Strategize, Explore and Define respectively, which constituted around 50% of the total variance. Desoete and Roeyers (2002) also found metacognition to be multidimensional construct; they further added that this will enable learners to adjust to varying tasks, demands and contexts. At the same time Boekaerts (1999) cautions us about metacognition, which is often used in an over, inclusive way, including motivational and affective constructs. Limitations: Validity and Reliability Statement of the Null and Alternative Hypothesis The null hypothesis for this research study would be, "Students with higher levels of success rate in solving calculus problems do not indulge in metacongnitive thinking strategies." Results & Discussions Observations Data was collected on students' perceptions towards metacognitive thinking by developing 48 different items. The inter-variable relationships were examined using Barlett's Test of Sphericity which indicated that the inter-variable relationship was statistically significant (l (435) = 4402.452, p = 0.001).The Kaiser-Meyer-Olkin (KMO) measure of the overall sampling adequacy (MSA) was 0.903, which demonstrated the strong inter-correlation among the items. The individual MSA scores indicated inter correlation within the items to be ranged from 0.840 to 0.941.This result suggested the presence of a high degree of correlation among the items and indicated the appropriateness for the usage of factor analysis. The measures for commonality of items presented scores ranging from 0.50 and above. The results of the above statistical tests have successfully established the suitability for running PCA. The result of PCA extracted 30 components out of the given items (Appendix 1). However, seven components with exigent values greater than 1 were retained for the study. The results tabulated for the seven components explained 59.53% of the cumulative variance in the data. The data set generated all of the seven hypothesized factors. Hence, the researcher has identified these seven components as responsible for students' metacognition thinking behavior towards problem solving. These seven components were identified as Self-efficacy, Exploration, Definition, Accommodation, Strategy, Execution and Verification. The second most significant predictor variable is Self-efficacy. In the past, various researchers such as Hackett, (1985); Pajares, (1996); Pajares & Miller, (1994) have reported that students' judgment of their capability to solve mathematical problems are indicative of their actual capability to solve novel and non-routine problems. Self-efficacy in mathematics also has been shown to be a strong predictor for mathematical problem-solving capability (Pajares.& Kranzler, 1995). Analysis These seven components explained around 60% of the total variance. Among these the higher variances were explained by Accommodation, Strategy, Exploration and Definition respectively, which constituted around 50% of the total variance. Desoete and Roeyers (2002) have also found metacognition to be amultidimensional construct. They further added that this will enable learners to adjust to varying tasks, demands and contexts. However, in his study Boekaerts (1999) has laid down cautionary standpoint about metacognition which asserts that it is often used in an over inclusive way, thereby including motivational and affective constructs. Research studies conducted by Collins (1982) and Schunk (1989, 1991) have reported that, when students approach academic tasks, those with higher self-efficacy are observed to work harder and for longer periods of time as compared to those with lower self-efficacy. Simultaneously, the enjoyment level experienced by the students while learning mathematics is directly concurrent with achievement of high scores. Apart from these traces, personal beliefs have also shown to have an influence on academic achievement (Garofalo, 1989; Kloosterman, 1995; Schoenfeld, 1985). However, the study conducted by Young and Ley (2002) reminds us that self-efficacy is not the only influence on achievement behavior. Higher self-efficacy will not produce competent performance when requisite knowledge and skills are lacking. Accommodation was calculated to be the third most significant predictor variable. It establishes that, it is one of the most central qualitative and subjective period for solving a non-routine or a novel problem. This is where the student puts together his interpretation of the question and his knowledge about the subject domain. At this point the student has completed the process of understanding the question and its connection to the subject domain. At this level of the operation, a student would have an exact understanding about the question and would be clear that what one can possibly do with the information provided within the question. Most importantly, the student would come to the stark clarity as in what he or she is supposed to calculate or find. Conclusions The research results indicate that the 'Verification' predictor is statistically insignificant. This component was intended to measure the student's ability to monitor his or her own strategy or work. The technique of solution monitoring allows an individual to analyze his problematic area so that he can come up with resolution measures which intend the construction and evaluation to resolve the problematic area for the effectiveness of the assorted procedures. The Verification indicator includes an individual's control over the internal representations he or she has formed and continues to form for the understanding and solving of a problem. This can be explained by the common observation that often, new strategies need to be formulated as a person realizes that the old ones have failed or not been up to a certain mark. This study has proficiently demonstrated that self efficacy is a predominant factor in the problem solving achievement of a student. Lecturers and particularly teachers should take heed of this discovery. Students should be psychologically prepared particularly at the younger age, to escalate their efficacy for mathematics and in particular problem solving. Teachers and lecturers should be encouraged to allocate significant amount of their time to promote the desire for acquiring their subject matter. This will tantalize the students' interest and at the same time motivate the students to acquire these information and skills voluntarily. Mathematics lecturers must participate in building up the cognitive construct from the inside out. Lecturers must embrace the view that they are responsible to bring the students to mathematics rather then to bring mathematics down to the students. In other words, students should be prepared to participate in this infinite journey of mathematics rather than just diluting the ideas and its functionality to barely meet surface level current demands. A student must be brought to the awareness that lecturers are systematically working out a specific problem rather then reproducing the answer from memory. Likewise, lecturers should prepare a manner of sort checklist for the students at the beginning of a lesson. Students should be then monitored and given reward points for organizing their thoughts in congruence with the provided checklist. This checklist should include the mechanism of metacognitive thinking strategies that have been discussed above. The study predicts that by adopting this mode of learning, at an early stage with continuous repetition over longer periods of time would suffice a more efficient classroom learning process. References Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal/or Research in Mathematics Education, 16, 163-176. Hackett, G. (1985). The Role of Mathematics Self-Efficacy In The Choice Of Math-Related Majors Of College Women And Men: A Path Analysis. Journal o/Counseling Psychology, 32,4756. Halpern, D. F. (1996). Thought and Knowledge: An Introduction to Critical Thinking. Mahwah, NJ: Lawrence Erlbaum Associates. Kloosterman, P. (1995). Students' Beliefs about Knowing and Learning Mathematics: Implications/or Motivation. Cresskill, N.J.: Hampton Press. Kluwe R. H. ( 1982). Cognitive Knowledge And Executive Control: Metacognition, Animal mind - human mind (pp. 201-224). New York: Springer-Verlag. Lawson, H. A. (1990). Beyond Positivism: Research, Practice, And Undergraduate Professional Education. Quest. 42, 161-183. Linn, M. M. (1987). Effects of Journal Writing on Thinking Skills of High School Geometry Students. University of Northern Florida: Masters of Education Project. Merseth, K. K. (1993). How Old Is the Shepherd An Essay about Mathematics Education. Phi Delta Kappan, 74(7), 548-592 Miller, M. D., & Pajares, F. (1997). Mathematics Self-Efficacy and Mathematical Problem Solving: Implications of Using Different Forms of Assessment. Journal 0/Experimental Education, 65(3),213-228. Owen, R. L., & Fuchs, L. S. (2002). Mathematical Problem-Solving Strategy Instruction for Third-Grade Students with Learning Disabilities. Remedial and Special Education, 23(5), 268-281. Pajares, F. (1996). Self-Efficacy Beliefs in Academic Settings. Review o/Educational Research, 66,543-578 Pajares, F., & Kranzler, J. (1995). Self-Efficacy Beliefs And General Mental Ability In Mathematical Problem-Solving. Contemporary Educational Psychology, 20,426-443. Pajares, F., & Miller, M. D. (1995). Mathematics Self-Efficacy And Mathematical Performances: The Need For Specificity Of Assessment. Journal o/Counseling Psychology, 42, 190-198. Paris S. G., & Winograd P. ( 1990). How Metacognition Can Promote Academic Learning And Instruction. Dimensions o/thinking and cognitive instruction (pp. 15-51). Peterson, P. L. (1988). Teachers' And Students' Cognitional Knowledge For Classroom Teaching And Learning. Educational Researcher, 17(S), 5-14. Philipp, R. A. (1996, November). Multicultural Mathematics and Alternative Algorithms. Teaching Children Mathematics, 3, 128-130. Reimann, P., & Schult, T. 1. (1996). Turning Examples into Cases: Acquiring Knowledge Structures for Analogical Problem Solving. Educational Psychologist, 31(2), 123-132. Rickard, A. (1995). Teaching with Problem-Oriented Curricula: a Case Study of Middle-School Mathematics Instruction. Journal o/Experimental Education, 64(1), 3-26. Schoenfeld, A. H. (1985a). Mathematical Problem-Solving. Hillsdale, NJ: Lawrence Erlbaum. Schoenfeld, A. H. (1992). Learning To Think Mathematically: Problem-Solving, Metacognition, And Sense-Making In Mathematics. Handbook on research on mathematics teaching & learning. New York: Macmillan Publishing Company. Schunk, D. H. (1989). Self-Efficacy and Achievement Behaviors. Educational Psychology Review, 1, 173-208. 19 Read More