STUDENTS’ BELIEFS AND MATHEMATICS PERFORMANCE IN A PROCESS – ORIENTED GUIDED – INQUIRY LEARNING (POGIL) ENVIRONMENT Jenyliza J. Tan-Ucang1 and Denis A. Tan 2,*
ABSTRACT An investigation was made to determine the effect of a ProcessOriented Guided-Inquiry Learning (POGIL) environment in teaching Geometry on students’ mathematics performance and beliefs. A pretestposttest and descriptive-correlational research design were used to ascertain if a significant difference would occur in students’ performance after exposure to POGIL and to identify predictors of mathematics performance, respectively. Two intact classes of sophomores were used as respondents. Questionnaires on students’ mathematics beliefs, problem solving beliefs, and Schommer’s epistemological beliefs were floated before and after the intervention. Students took the pretest and posttest in Geometry, a researcher-made test, validated by experts and pilot tested with a Cronbach alpha of 0.80. Results show that respondents believed most on simple knowledge and the least on fixed ability. Also, they had moderately refined conception on the nature of Mathematics prior to and after POGIL instruction. T-test of difference indicated that their performance increased significantly. Significant correlation between their performance and beliefs was established using Pearson product moment correlation. Linear regression analysis revealed that students’ epistemological beliefs is the best predictor of mathematics performance. Other factor, like nature of math belief, also influenced their performance. Findings show that both general and domain specific beliefs affected students’ performance. Keywords: P O G I L , performance, beliefs, cooperative learning 1
Faculty, Professional Education Department, College of Education, Central Mindanao University, Bukidnon, Philippines ([email protected]
) 2,* School Director, CMU Laboratory High School, Faculty, Professional Education Department, College of Education, Central Mindanao University, Bukidnon, Philippines (Corresponding Author: [email protected]
INTRODUCTION Many researches have been conducted to reveal the beliefs of the teachers and their epistemological commitment, as well as its effects on their classroom instructional practices. But, only few studies were undertaken to determine the effect of these beliefs on the students’ learning (Roth & Roychoudhury, 1994). On the other hand, it seems rational to presume that the belief of the students will influence their point of view toward the activities in the classroom and their academic performance.
The understanding of the students on the nature of knowledge and the learning strategies that they consequently used have been developed throughout their stay at school. The way that the nature of knowledge is presented will affect students’ understanding to the subject, and as a result, how they relate to knowledge is affected (Roth & Roychoudhury, 1994). If mathematics is offered and presented as an organization of proven data, rigid truth, relationships, and procedures to students, then they may think that learning the subject is more on memorizing formula, and they will presume that concepts in mathematics can be best understood if one knows the algorithm in finding the correct answer. This line of thinking is a behaviorist in nature. Students’ exposure to this kind of environment (behaviorist) will also greatly affect their beliefs on learning mathematics. No doubt that most o f t h e students’ mathematics beliefs are naïve and still in the behaviorist framework. Schoenfeld (1989) argued that as a result of direct instruction, many students develop some beliefs about “what mathematics is all about” that are plainly wrong – and that those beliefs have a very strong negative effect on the students’ mathematical behavior.” This seems to suggest that changing students’ experience about learning mathematics may change their beliefs regarding the subject. The trend in education now is to help teachers change their epistemology of teaching from objectivist view to that of a constructivist. Discrepancies in the classroom environment may happen by changing only t h e teachers’ views. Thus, students’ epistemological beliefs should be understood. Thus, it is necessary to find out students’ epistemological beliefs to understand their perspective in knowing and learning mathematics. Teachers’ knowledge on students’ beliefs will help them design activities that will promote changes on those naïve beliefs. Furthermore, it will also help in designing a new learning environment that will promote mathematical thinking among students that is suited to their level.
Beliefs, as described by Kloosterman and Cougan (1994), are significant influences on the behavior of every human being. Every individual, when faced with a choice of actions, will do actions that will upshot in consequences that they want (Bartsch & Wellman, 1989). A number of studies had investigated students’ beliefs and its relation to academic performance. Schommer-Aikins (2002) examined the structure of middle schools students and found out that the belief systems, such as epistemological beliefs and mathematical problem solving beliefs, predict academic performance of students. According to Pajares (1992), beliefs are dominant contributors than knowledge in distinguishing the organization and definition of individuals on their tasks and problems since they are stronger predictors of behavior. This is supported by Navaja (2005) when she reiterated that the relationships between beliefs and actions is interactive. These studies suggest that beliefs of students influence their actions and behavior. This means that their performance is influenced by these beliefs. Educational beliefs are “broad and encompassing” (Navaja, 2005). According to Hofer and Pintrich (2002), the way students attack the material and what they understand may significantly be affected by their beliefs and knowledge about knowledge and learning. These epistemological beliefs have been linked to numerous aspects of academic learning, in particular, and the college and high school students (Schommer-Aikins, Mau, Brookhart, & Hutter, 2000). For instance, when college students less likely believe in the simplicity of knowledge, the more they exert effort to comprehend the academic text, monitor their comprehension, and use sophisticated study strategies (Schommer, 1990; Schommer, Crouse, & Rhodes, 1992; Schraw, Dunkle, & Bendixen, 1995). Also, the high school students do not give more value to education if they believe in the innate ability to learn (Schommer & Walker, 1997). Since 1990, it was found out that numerous aspects of academic performance of high school students are being predicted by the epistemological beliefs. Understanding, metacomprehension, and analysis of information among college students can be predicted by the beliefs in structure and certainty of knowledge (Schraw et al., 1995). Comprehension, valuing education, and overall grade point average for college students can also be predicted by beliefs about the speed of learning and ability to learn (Schommer & Walker, 1997). Strategies used by students in studying and comprehension of complex text are related to beliefs in simple knowledge (Schommer et al., 1992).
Solving a problem in a not so well-structured content is correlated to students’ belief on the certainty and simplicity of knowledge (Schraw et al., 1995). Conviction on quick learning predicted problem solving in a well-structured content (Schraw et. al, 1995). When high school students perceive that learning is quick, then they likely earn a low grade point average. Schommer-Aikins (2002) studied the structure of middle school students’ general beliefs by asking whether st ud e nt s ’ acad e mi c p er fo r ma nc e ca n b e p red ic ted b y the two belief systems (epistemological beliefs and domain-specific mathematical problemsolving beliefs). Findings of the study revealed that beliefs about useful math, effortful math, understand math concepts, and math confidence are significantly related to beliefs in quick/fixed learning and studying. In addition, the result in path analysis suggested that academic performance as measured by solving mathematics problems and overall grade point average can be predicted by both general and domain specific epistemological beliefs. Kloosterman and Cougan (1994) found out that increasing beliefs that result in high motivation would result in high achievement among respondents. Schoenfeld (1985) had studied gifted students’ successful problem-solving skills, and he concluded that students’ beliefs about the nature of mathematical knowledge and learning influence some parts of the student problem-solving process. Also, Duell and Aikins (2001) said that the nature of science had little correlation between beliefs and reasoning. They established that those students who showed no increase in t h e i r epistemological beliefs exhibit strong advancement in rational thinking. This indicated that not all students who had changed their behavior had also changed their beliefs. NCTM (1987) revealed that a non-linear correlation between beliefs and learning occurred. Students’ beliefs on how it takes to learn mathematics are likely influenced by their learning experiences. Sequentially, the students’ approach to new mathematical practices will likely be influenced by their beliefs about the subject. The literature suggested that mathematics’ beliefs of students who are gifted and might be average are never different. The beliefs of students on mathematics learning environment differ, depending on what the teacher would be using inside the classroom and their domain general epistemological beliefs. Students’ beliefs are stable when not challenged. The types of beliefs that the students possess depend on the classroom orientation that they have, their personal experience with mathematics inside the classroom, and their beliefs on knowledge and knowing. 144
Students’ orientation and exposure to mathematics activities influence their beliefs in the subject (Tan, 2009). This exposure to the teaching that emphasizes mastery of skills, memorization of formulas, and arriving at the correct answer, prompted them to perceive that Mathematics is formulaic, structured, and memorization. In addition, if students’ experience of Mathematics in a class is quite behaviorist, later in their lives, they will believe that Mathematics is merely finding the correct answers, following predetermined procedure by the teacher in solving problems, and memorizing formulas and more on algorithm. Process-Oriented Guided-Inquiry Learning (POGIL) is a laboratory and instructional technique wherein the students work collaboratively as a team. It seeks to teach instantaneously the content and main process skills to the students that will enhance students’ ability to think analytically and work effectively in a group. A POGIL classroom or laboratory is designed in such a way that the students working in small groups will be guided by the inquiry material that will be given to them. These materials serve as guide, and the data and information followed by leading questions that are needed by the students are there to help the students toward formulation of their own valid conclusions. In the POGIL classroom, the students will be observed periodically so that their needs will be addressed properly since the instructor serves as a facilitator of learning only. POGIL is a n o u t p u t o f a r e s e a r c h indicating that a) m o s t s t u d e n t s a r e n o t m o t i v a t e d b y teaching o r by telling m e t h o d ; b) most successful students are those who are part of a collaborative community; c) when students are given an opportunity to construct their own understanding; and d) students enjoy themselves more and improve better proprietorship over the material because knowledge becomes individual. A discovery-based team environment provides teachers with constant feedback about what their students comprehend and misconstrue, and it also energize the students. The discovery-based team environment that develops among students will help them value their logical thinking and teamwork than simply getting "the correct answer." Thus, it emphasizes that learning is a cooperative process of filtering understanding and developing skills, and not merely a task of remembering information (Hanson & Moog, 2006).
POGIL’s efficiency has been evaluated by many institutions to diversified courses (Farrell, Moog, & Spencer, 1999; Hanson & Wolfskill, 2000). Numerous collective and significant findings of the studies revealed that: a) students are more focused in POGIL than customary methods; b) generally, students have advanced mastery of content for POGIL compared to customary methods; c) and POGIL is mostly preferred by students over other methods. In the POGIL, students learn best when they are engaged actively in the analysis of models; when they discuss ideas; when they work together; when they reflect on what they have learned and think about how to improve performance; and when they interact with an instructor. POGIL utilizes self- managed teams, guided-inquiry materials, and metacognition to support the research-based learning environment (Hanson & Moog, 2006). This study made use of this new learning environment in teaching mathematics. It is hoped that POGIL would enhance learning among students as they work in teams and have a paradigm shift in mathematics learning. Objectives of the Study The study endeavored to find out the usefulness of POGIL in teaching Geometry among sophomore students. Specifically, it aimed to describe the demographic profile of the students based on their age, gender, type of elementary school where they came from, and parents’ educational attainment. It also aimed to ascertain the level of students’ beliefs before and after exposure to POGIL instruction in terms of epistemological beliefs, and mathematics beliefs; differentiate the performance of students before and after POGIL instruction; correlate students’ performance and beliefs in terms of epistemological beliefs and mathematics beliefs; and lastly, determine the best predictor of mathematics performance. METHODOLOGY A quasi-experimental research design was used to establish if a significant difference in the performance of students would occur after exposure to POGIL. The respondents of the study were the Central Mindanao University High School (CMULHS) sophomore students who took Geometry during the school year 2011–2012. Both the whole group of the first two sections, namely Gemini and Virgo, were used as respondents of the study.
Questionnaires were administered to all participants during their regular classes inside their classrooms. Confidentiality of their answers were observed. Pre-test and posttest on mathematics belief scale, Schommer’s Epistemological Beliefs questionnaire (adopted with permission from the author), and teacher-made Geometry test were the instruments of the study. The researcher-made test was validated by experts and reliability was established and found to be 0.80. The rest of the instruments were adopted and were pilot-tested with Cronbach alpha ranges from 0.63 to 0.85. Descriptive statistics, Spearman rank, multiple regression, and t-test were used in the data analysis to answer each research problem. In the interpretation of epistemological beliefs, the Schommer’s interpretation of data was used. RESULTS AND DISCUSSION The presentation of the results in this investigation is arranged parallel to the numbering of the statement of the problems. In Table 1, majority of the respondents were 13 years old. Out of the 93 respondents, 16 (17%) were 12 years old, 53 (57%) were 13 years old, and 24 (26%) were 14 years old. Also, it was found out that the majority of the respondents were female. Most of them came from public elementary schools. For the respondents’ parents’ educational attainment, the majority were children of college level or college graduate parents. This implies that the respondents were children of educated parents. All respondents were given a 63-item questionnaire to determine their epistemological beliefs. The respondents’ responses in each question were categorized into twelve subcategories of epistemological beliefs based on Schommer’s criteria. Then, these s u b c a t e g o r i e s we r e analyzed and finally trimmed down into four dimensions.
Table 1. Students Demographic Profile Variables
12 13 14
16 53 24
17 57 26
Elementary School type
Elementary Level and Elementary Graduate
High school level and high school graduate
College level and college graduate
Elementary level and elementary graduate
High school level and high school graduate
College level and college graduate
Father’s Educational attainment
Mother’s Educational attainment
Table 2 shows the summary of the analysis of students’ epistemological beliefs based on Schommer’s tool and interpretation. The findings revealed that students believed most on simple knowledge which has the highest mean of 3.346 before POGIL instruction.
This means that students tend to avoid ambiguity, seek single answers, avoid integration, and depend on the authority before the conduct of the study. However, after the investigation, the belief of the respondents remained the same, although there is a slight increase in the number. Table 2. Students Epistemological Beliefs Based on Schommer’s Criteria Beliefs
Pre-test (Mean Score)
Post-test (Mean Score)
Similarly, among the four dimensions, respondents believed the least on fixed ability before exposure to POGIL environment. This means that they still cannot learn how to learn, they do not attribute success to working hard, and they believe on innate ability and learning things for the first time. Still, respondents perceived the aforementioned concepts the least after the investigation. These findings on students’ stable epistemological beliefs support the study of Duell and Aikins (2001) when they found out that those students who have no gains in their beliefs exhibited enhanced growth in their rational thinking. Table 3 presents the levels of mathematics beliefs of the respondents. Among the 15 statements, respondents answered “there are many ways to get the correct answer in Mathematics problems” with the highest mean of 3.71 in the pretest and 3.76 in the posttest. This means that respondents have sophisticated beliefs on the nature of mathematics on this item. Also, they have refined conception on the connectedness of Mathematics to real life. This indicates that students view mathematics learning in this item in a constructivist perspective. Respondents had slight changes on their beliefs on the nature of Mathematics after POGIL instruction. But in terms of its interpretation, they fall on the same category. Respondents have very naïve beliefs on the following items: learning Mathematics is simply memorizing and mastering formulas; learning Mathematics is simply memorizing and mastering formulas; in learning to do mathematics, there is always a rule to follow; doing Mathematics is to follow rules set by the teacher; knowing Mathematics 149
means to remember and apply the correct rule; and mathematical truth will be determined when the teacher confirms the answer. These indicate that respondents have behaviorist views on mathematics in these items. On the other hand, they had refined conceptions on the rest of the items. These are basically true to most of the students not only in CMULHS. In totality, the respondents had moderately refined conception on the nature of Mathematics before and after exposure to POGIL with a mean of 2.75 and 2.77, respectively. There is a slight increase of the over -all mean score of the respondents. However, this does not guarantee the change in their beliefs. The findings are supported by NCTM (1987) when they emphasized that students’ beliefs are stable when not challenged. Although POGIL may challenge their naïve beliefs, the time of exposure to this new environment may not be enough to influence the belief of students. Table 4 shows the changes on the performance of students before and after instruction. Out of 93 students who took the exam, the mean score in the pretest is 7.66 and in the posttest is 9.91. T- test revealed that there is a significant increase in the performance of the students as reflected by its tvalue of -6.714 with p-value of 0.000. This indicates that POGIL instruction is effective in affecting change in performance of students in Mathematics. This is supported by the study of Hanson and Wolfskill (2000) when they found out that students exposed to POGIL generally have sophisticated mastery of subject matter than those students exposed to customary methods, and it follows that most students favor POGIL over other methods. The significant change in the students’ performance may be caused by the learning environment that is competitive, individualized, or cooperative. Essential process skills (e.g. critical and analytical thinking, problem solving, teamwork, and communication) were acquired by the students working in a team environment (Hanson & Moog, 2006).
Table 3. Students’ Mathematics Beliefs STATEMENTS
1. Learning Mathematics is simply memorizing and mastering formulas.*
6. Knowing Mathematics means to remember and apply the correct rule.*
7. Mathematical truth will be determined when the teacher confirms the answer.*
10. Learning mathematics is more on understanding patterns rather than memorizing formulas.
11. Doing mathematics is more on seeking solutions, not just memorizing procedure.
12. Learning mathematics is more on formulating conjectures, not just doing exercises.
14. There are many ways to get the correct answer in solving mathematics problems.
15. Mathematics learning is fun and challenging.
2. Classroom mathematics is not related to the outside world.* 3. School Mathematics is mainly mastering procedures.* 4. In learning to do mathematics, there is always a rule to follow.* 5. Doing Mathematics is to follow rules set by the teacher.*
8. There is one right method in solving problems.* 9. Mathematics is related to real life.
13. Mathematics is an exploratory, dynamic and evolving discipline.
Over-all Mean * Reversely scored
4.50 – 5.00
3.50 – 4.49 2.50 – 3.49
- Highly Sophisticated -Sophisticated
1.50 – 2.49
-Moderately Refined -Naïve
0.00 – 1.49
-S -MR -N -VN
Table 4. Students Performance before and after POGIL instruction N
α = .01 Level of Significance
Table 5 reveals the relationship between students’ performance and beliefs. Findings indicated that performance is significantly and negatively correlated with beliefs on ‘certain knowledge’ with r-value of -2.202. This means that when students got higher score in Geometry test, they believed least on the certainty of knowledge. On the other hand, those students whose score were low, strongly believed that knowledge is certain. This finding is supported by the study of Duell and Aikins (2001) when they found that epistemological beliefs have little correlation on the measure of learning. Also, it was established that a low correlation existed between reasoning and beliefs. Students who exhibited strong improvement in reasoning may show no change in their epistemological beliefs. In addition, oftentimes, the correlations between epistemological beliefs and measures of learning and reasoning are relatively low. Table 5. Correlation between Students’ Performance and Beliefs BELIEFS
Mathematical Beliefs Nature of Math
On the other hand, performance of students is positively correlated with their beliefs on the nature of Mathematics. This indicates that the more sophisticated or refined is the conception of students on the nature of Mathematics, the higher is their performance in M a t h e m a t i c s and vice versa. This finding supports the claims of Kloosterman and Cougan (1994) that increasing beliefs that result in high motivation will result in high achievement. Table 6 presents the result of stepwise multiple regression of the independent variables and performance in Geometry. It was revealed that 31.1% of students’ performance in Geometry can be accounted for by the belief on the nature of Mathematics and the epistemological beliefs. These beliefs have higher beta weights of 3.039 compared to beliefs on the nature of Mathematics which is 2.571. This indicates that epistemological beliefs is the best predictor of performance of students in Geometry. Table 6. Multiple Regression Analysis of the predictor variables of Academic Achievement
(Constant) Epistemological beliefs (Certain
Unstandardized Coefficients B
Standardized Coefficients t
Knowledge) Nature of math beliefs R= 0.557
R squared = 0.311
Sig. = 0.000
As explained earlier, the coefficient of determination R2 = 0.311 suggests that 31.1% of the variation in students’ performance in Geometry can be explained by the combined effect of the aforementioned variables. On the other hand, the 68.9% can be explained by the variables not included in the study such as attitude of the students, their computational skills, their study habits, their IQ, and many others. 153
The belief of the respondents on the certainty of knowledge is the best predictor of their performance. The beta weight of 0.242 indicates that for every increase of one standard deviation unit of students’ belief on the certainty of knowledge, the performance of students in Mathematics would also increase in 0.242 standard deviation unit, with all the other independent variables held constant. On the other hand, belief on the nature of Mathematics with a beta weight of 0.196 means that it is the second best predictor of performance. This indicates further that for every one standard deviation unit increase on students’ belief on the nature of Mathematics, there would be a 0.196 standard deviation unit increase on students’ performance. Thus, the regression equation model for students’ performance in Geometry in this study is the following: Y’ = 13.502 + 3.039X1 + 2.571X2 Where :
Y’ = Performance in Mathematics, X1= Epistemological Beliefs (Certain Knowledge), X2= Beliefs on the nature of Mathematics
The results of this investigation sustain NCTM’s (1987) notion that assessment of students’ mathematical beliefs is an important component of the overall mathematical knowledge performance of students. In addition, results of this s t u d y validate NCTM’s claim that a dominant predictor of students’ assessment on their capabilities, on their initiative to take part in a mathematical activity, and on their mathematical outlook is their belief. Results of this study also confirm the findings of Schommer-Aikins (2002) that epistemological beliefs and beliefs on the nature of Mathematics, respectively, predicted academic performance. CONCLUSIONS AND RECOMMENDATIONS Based on the findings of the researchers’ investigation, they concluded the following:
Sophomore students are dominated by females. Two thirds of the students came from the public schools, and one third of them came from the private elementary schools. Majority of their parents are professionals. Prior to and after POGIL instruction, epistemological belief on simple knowledge got the highest mean, while belief on fixed ability has the lowest mean. On the other hand, their mathematical belief is moderately refined. Students’ scores are not that high prior to POGIL instruction and scores increase after the intervention. 154
There is a significant increase in the students’ performance before and after POGIL instruction. Students’ performance is negatively correlated to epistemological belief on the certainty of knowledge while positively correlated to mathematical belief specifically on the nature of Mathematics. Epistemological belief and belief on the nature of Mathematics are the predictors of mathematics performance. Between the two predictors, epistemological belief is found to be the best predictor of performance. Through these conclusions, the researchers recommended that a longer period of time is needed in the conduct of POGIL instruction to identify the change in the epistemological belief and mathematical beliefs of the students. It will also ascertain significant change if there is any. Other factors affecting students’ performance should also be looked into, not only the age, parents’ educational attainment, and the type of school where they graduated. Further study should be conducted to identify other factors that predict mathematics performance of the students.
LITERATURE CITED Bartsch, K. & Wellman, H. (1989). Young children’s attributes of actions to beliefs and desires. Child Development, 60, 946-964. Duell, O.K., & Schommer-Aikins, M. (2001). Measures of people’s belief about knowledge and learning. Educational Psychology Review, 13, 419-449. Farrell, J. J., Moog, R. S., & Spencer, J. N. (1999). A guided inquiry general chemistry course. Journal of Chemical Education, 76, 570-574. Hanson, D.M. & Moog, R. S. (2006). Introduction to POGIL. Retrieved from http://www.pcrest.com/PC/pub/POGIL.htm Hanson, D., & Wolfskill, T. (2000). Process workshops - A new model for instruction. Journal of Chemical Education, 77, 120-130. Hofer, B. K., & Pintrich, P. R. (Eds.). (2002). Personal epistemology: The psychology of beliefs about knowledge and knowing. Mahwah, NJ: Lawrence Erlbaum Associates. Kloosterman, P., & Cougan, M.C. (1994). Students’ beliefs about learning school mathematics. The Elementary School Journal, 94(4), 375 -388. National Council of Teachers in Mathematics (1987). Principles and standards in mathematics. Retrieved from http://www.nctm.org/ Navaja, L.M. (2005). Changing pre-service science teachers’ beliefs through constructivist technology-aided learning (Unpublished doctoral dissertation). De La Salle, Manila. Pajares, M.F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62, 307 – 332. Roth, W.M., & Roychoudhury, A. (1994). Physics students’ epistemologies and views about knowing and learning. Journal of Research in Science Teaching, 31, 5-30. Schoenfeld, A. H. (1985). Mathematical problem-solving. New York: Academic Press. Schoenfeld, A. H. (1989). Explorations of students’ mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20, 338-355. Schommer, M. (1990). The effects of beliefs about the nature of knowledge on comprehension. Journal of Educational Psychology, 82, 498-504. 156
Schommer, M., Crouse, A., & Rhodes, N. (1992). Epistemological beliefs and mathematical text comprehension: Believing it is simple does not make it so. Journal of Educational Psychology, 84, 435443. Schommer, M., & Walker, K. (1997). Epistemological beliefs and valuing school: Considerations for college admissions and retentions. Research in Higher Education, 38, 173-186. Schommer-Aikins, M., Mau, W., Brookhart, S., & Hutter, R. (2000). Understanding middle students’ beliefs about knowledge and learning using a multidimensional paradigm. Journal of Educational Research, 94, 120-127. Schommer-Aikins, M. (2002). An evolving theoretical framework for an epistemological belief system. In B.K. Hofer, & P.R. Pintrich (Eds.), Personal epistemology: The psychology of beliefs about knowledge and knowing (pp. 103-118). Mahwah, NJ: Erlbaum. Schraw, G.S., Dunkle, M.E., & Bendixen, L.D. (1995). Cognitive processes in well-defined and ill-defined problem-solving. A pplied Cognitive Psychology, 9, 523-538. Tan, D (2009). Using metacognitive skills in the mathematical problem solving heuristics among senior high school students (Unpublished doctoral dissertation). De La Salle University, Manila.
Acknowledgement: The r esearchers would like to extend their heartfelt gratitude to the University President, Dr. Maria Luisa R. Soliven, the University Vice President for Research and Extension, Dr. Luzviminda T. Simborio, the Director of Research, Dr. Maria Estela B. Detalla, the Research Coordinator, Dr. Andrea Azuelo, and the Dean of the College of Education, Dr. Raul C. Orongan, for the research grant. Also, they would like to extend their thanks to the High School Director, Dr. Denis A. Tan, the High School students for the help extended during the data gathering phase of this study. Above all, to the Almighty God the Father, whom with Him, everything is possible.