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Proceedings of the 3rd International Conference of Teaching and Learning (ICTL 2011) INTI International University, Malaysia 1 QUALITATIVE REASONING APPROACH FOR IMPROVED CONCEPTUAL UNDERSTANDING AND PROBLEM SOLVING PERFORMANCE IN STATISTICS Edna C. Aquino1, Luisito Hagos2, Susan E. Puyat3, Virginia S. Sobremisana4 and Lorena M. Halili5 1,2,3,4 Rizal Technological University, Philippines; 2New Era University, Philippines (2dochagosneu@yahoo.com, 3susan.puyat@yahoo.com.ph, 5ringgai_30@yahoo.com ul) ABSTRACT Guided by the conceptual change theory, this study was conducted to determine the effect of the Qualitative Reasoning Approach (QRA) in the conceptual understanding and problem solving performance in Statistics. The Qualitative Reasoning Approach (QRA) is a problem solving strategy that emphasized both quantitative and qualitative aspects of problem solving in Statistics. Analysis of the data gathered indicated that the QRA in solving worded problems significantly improved not only the students’ conceptual understanding but their problem solving performance. KEYWORDS Statistics, Performance, Qualitative Reasoning, Problem Solving, Conceptual Understanding, Understanding. Reasoning, Qualitative Reasoning Approach INTRODUCTION The concept of education has changed from the traditional subject-centered classroom to the more innovative learner-centered classroom. This transition therefore encourages teachers to create situations where students can be active, creative, and responsive to the physical world. This can be done by asking students to explore, justify, represent, discuss, use, describe, investigate, predict; in short, by being active in the learning process. The goal is for students to see that Statistics is practical and used daily with the hopes of students acquiring a deeper appreciation the subject. Teachers are also encouraged to become instruments in the development of higher order thinking skills (HOTS) through active involvement in learning mathematical content. This viewpoint is also reflected in the new standards for mathematics instruction from the National Council of Teachers of Mathematics (1989, 1991, 2000, NCTM). The NCTM standards propose new expectations for what students should learn and about how this learning should occur. They mandate increase emphasis upon complex problem solving, higher level reasoning, and making connections across mathematical domains, and communication. In addition, they Proceedings of the 3rd International Conference of Teaching and Learning (ICTL 2011) INTI International University, Malaysia 2 recommend decreased emphasis upon rote computation, routine problem solving, and overdependence upon the teacher or text for explaining and evaluating concepts and procedures. In short, instruction should de-emphasize the transmission of factual or otherwise incontestable information from teachers to students and should encourage the active involvement of students in discussing ideas, making convincing arguments, reflecting on, and clarifying their thinking. Recognizing the vital role played by the teachers in initiating meaningful reforms in the Statistics classroom, studies must therefore be done to develop methods and strategies that will eventually improve the achievement of the students. Hence, this study focused on the effect of Qualitative Reasoning Approach (QRA) on students’ conceptual understanding and problem-solving performance in statistics in the tertiary level. Specifically, this study was designed to determine if the students’ conceptual understanding and problem solving performance had significantly improved after their exposure to the qualitative reasoning approach. The main purpose of this study was to investigate the effect of the Qualitative Reasoning Approach (QRA) on students’ conceptual understanding of statistics in terms of Statistics Concepts Inventory (SCI) score. It also identified the students’ problem solving performance using the problem solving test. THEORETICAL FRAMEWORK The main theoretical framework of this study is the theory of conceptual change. The theory posits that students learn when an existing conception that they already possess is challenged. Conceptual change theory is based on Jean Piaget’s ideas and writings on how students learn. Conceptual change theory has been a major theoretical framework for science education, especially physics in the last thirty years or so. It has only been transferred to other fields in a limited capacity. However there are a few recent studies in mathematics and statistics that have used conceptual change theory successfully as their theoretical framework (Crisostomo, 2004). In conceptual change, an existing conception is fundamentally changed or even replaced.. Various studies in science education have shown that when students come to conceptual understanding of a topic through conceptual change instruction, their conceptual skills are much stronger because they understand how they come to their conceptual understanding (Vosniadou, 1994). Proceedings of the 3rd International Conference of Teaching and Learning (ICTL 2011) INTI International University, Malaysia 3 INSTRUCTIONAL STRATEGIES Teachers play a major part in the learning process. It is through them that learners learn new concepts, knowledge and ideas essential to improve their performance. To facilitate learning, there is need for teachers to identify the needed teaching methods and techniques to improve students’ ability to solve problems and develop conceptual knowledge. Qualitative Reasoning Approach (QRA) The Qualitative Reasoning Approach (QRA) is a problem solving method used to solve worded problems. The QRA emphasizes not only the quantitative but also the qualitative aspect of problem solving. The strategy is presented by the teacher through problem solving activities giving particular attention to the procedures and the students are encouraged to apply the same procedures in their seat works, assignments, and examinations (Siguenza, 2006). The QRA in problem solving is an instruction that teaches the students how to solve worded problems using a set of general guidelines as shown in Figure 1. Figure 1. The Steps for Solving Statistics Problems Using the QRA The first step in QRA is to define known and unknown quantities. The known and unknown quantities are written, the condition, premise and limitations of the problem are stated. The second step is to select an equation. This involves writing the formula or mathematical relations that can be used to determine unknown quantities. The third step is to solve the given problem. This involves manipulating equations and substituting known quantities into the equation to solve for the unknown quantities. In this step, previous knowledge of mathematics is of great help for the students to simplify the relation. The next step is to check the answer. The unit of measure and the sign involved in the final answer is checked to confirm if it really satisfies the unknown quantity, the order of magnitude is reasonable, and what is asked in the problem has Proceedings of the 3rd International Conference of Teaching and Learning (ICTL 2011) INTI International University, Malaysia 4 been answered. The fifth and final step of the strategy is to write explanation and reasoning by presenting the problem situation in words, identifying concepts and principles used in reasoning and explaining the process followed in solving the unknown quantity. The first four steps are similar to the traditional textbook problem solving. The last step facilitates the more concrete qualitative aspect of problem solving. The written explanation and reasoning serve to help the students improve their problem solving performance and understand better the concepts and principles of statistics. Conceptual Understanding The major goal of Philippine education calls for the development of life skills necessary to prepare the Filipino child to successfully deal and adapt to the society where he is in existence. Along this notion, it is imperative to explore the factors that may improve or inhibit the growth of conceptual understanding. Understanding can be defined as the ability to use knowledge, to cope with situations. One’s knowledge of a topic may consist of sparsely connected elements, while the same number of elements in a person’s mind could be bound into a coherent mass by many shared terms. In short, understanding of a concept is not only a function of the extent of knowledge about it, but also of the integration of that knowledge (Huffman, 1997). Traditional mathematics teaching mainly cultivates skills, neglecting conceptual understanding of the underlying domain. The students’ learning difficulties in acquiring the concepts of mathematics is abstract nature of mathematics. Since mathematical concepts are abstract, students learn mathematics by memorizing. One of the most important problems associated with the teaching of mathematics rise from the students’ understanding difficulties in establishing the relationship between their knowledge and intuition about concrete structures and abstract nature of mathematics thus, making it not easy to find concrete examples in mathematical concepts. Also, mathematical concepts are abstract that one needs highly cognitive achievements to assimilate them. Problem-Solving The National Council of Teachers of Mathematics (NCTM) argued that problem solving should become the “focus” of mathematics in school and that centering mathematics instruction around problem solving can help all students learn key concepts and skills within motivating contexts (Lubienski, 2002). Problem solving in mathematics can be explained as “thinking and working mathematically” but the converse is not true. Problem solving in mathematics is an intricate process which calls for a problem solver who is engaged in a mathematical task to organize and deal with domain specific and domain-general pieces of knowledge (Yeo, 2009). Proceedings of the 3rd International Conference of Teaching and Learning (ICTL 2011) INTI International University, Malaysia 5 Lester and Kehle (2003) typify problem solving as an activity that involves the students’ engagement in a variety of cognitive actions including accessing and using previous knowledge and experience. Effect of QRA to Conceptual Understanding and Problem Solving Performance Since Statistics is an Applied Mathematics, its varied applications should focus on real life situations and as such, teaching problem solving that would entice the learners to come up with accurate, step by step solutions and applications of the various concepts learned is a task a Statistics teacher has to face. Aside from the QRA, other teaching strategies that elicit reasoning and explanation were also identified and they were proven to help improved conceptual understanding and problems solving performance. Statistics as a discipline plays an important role in almost all fields of human endeavor. As a branch of Mathematics, it contains abstract concepts that in one way or another hinder students’ learning and may decrease their desire to learn the subject. It is then the task of the educator to implement and design teaching strategies that will enable statistics students to better understand statistical concepts and eventually appreciate its application to real life situations. The Qualitative Reasoning Approach (QRA) to problem solving is a method of solving worded problems such that not only the problem solving skills are being enhanced but it also develops the student’s reasoning skills and the ability to verbalize the actions that took place during the course of problem solving. It is the aim of this study that by solving worded problems following the steps as suggested by the QRA culminating it with reasoning, the students will develop in themselves a better conceptual understanding of Statistics and perhaps would be able to see solving problem in a different light if not fully appreciate the beauty of Mathematics. METHODOLOGY The study utilized the quasi-experimental research design that approximates the conditions of the true experiment in a setting which does not allow the control and/or manipulation of all relevant variables. Quasi-experimental studies examine outcomes; however, they do not involve randomly assigning participants to control and treatment groups. This type of study compares outcomes for individuals before and after the group’s involvement in a program (www.childternds.org, 2008). Several types of quasi-experimental designs are identified but in this study, the pretest-posttest, nonequivalent control group design was utilized. This design looks a lot like the randomized pretest-posttest design, but in this case the two groups have not been equated prior to treatment. Since one has not randomly assigned subjects to groups, one cannot assume that the populations being compared are equivalent on all things prior to the treatment (Allen, 2006). Proceedings of the 3rd International Conference of Teaching and Learning (ICTL 2011) INTI International University, Malaysia 6 Intact classes which were matched were used as respondents of this study and they were not isolated so as to avoid detection that they were part of a study. Research Instruments In the conduct of the study, the researcher made use of the Statistics Concept Inventory (SCI) and Problem Solving Test (PST). Statistics Concepts Inventory (SCI) The Statistics Concept Inventory (SCI) is a concept inventory similar to the Force Concept Inventory (FCI) and other concept inventories, the ultimate goal of which is to increase understanding and awareness of what, why, and how students learn. The SCI as a whole can be said to measure the construct “statistics knowledge” or, more precisely, “conceptual understanding of statistics” (Allen, 2006). To be able to measure the conceptual understanding of the students using the SCI, the problem solving activities was conducted so as to let the students explore their problem solving techniques while letting them apply previous concepts learned and link it with new knowledge and ideas through the QRA. Problem Solving Test (PST) This researcher-made instrument consisted of a set of worded problems in Statistics similar to Statistics problems found on the textbook prescribed by the university. Each worded problem was constructed for the students to generate mathematical solution. Solutions to these problems will be used to describe as well as assess the students’ problem solving performance. The aforementioned test was administered as pretest and posttest to the experimental and control groups. The topics included in the problem solving test were taken from the course syllabus of Statistics I prescribed by the university. The topics included were Measures of Symmetry, Selected Topics in Probability, Normal Distribution and Correlational Techniques. The SCI was used to measure the conceptual understanding of the students, the problem solving test will further vouch the students’ problem solving performance in statistics. Table 1 shows the summary of the difference between the SCI and QRA. Proceedings of the 3rd International Conference of Teaching and Learning (ICTL 2011) INTI International University, Malaysia 7 Table 1. Summary of the Difference between the SCI and QRA Type Statistics Concept Inventory Qualitative Reasoning (SCI) Approach (QRA) Test Problem solving strategy Test Standardized Researcher-made Multiple choice test Procedural steps Category Method Usage Spectrum Scoring Measures conceptual Identifies problem solving understanding performance Quantity Quantity and Quality Counting the number of Scoring rubric correct responses FINDINGS The findings of the study are as follows: 1. The experimental group obtained an average score of 5.04 in the pretest while the control group has a mean score of 5.48. Both groups fall below achievement level with a marginal difference. 2. The posttest average scores in the problem solving activities of the experimental and control groups are 77.20 and 67.00 respectively. This showed a mean gain of 10.20. 3. The computed value of t is – 0.75, less than the critical value of t at .05 level of significance with 48 degrees of freedom at – 2.0126 when their absolute values are compared. Hence, no significant difference exists between the experimental and control groups when their pretest average scores in the SCI are concerned. 4. The posttest average scores yielded a t – computed value of 5.485 greater than the tabular value of 2.0126 and the null hypothesis was rejected. It implies a significant difference in the posttest average scores in the problem solving activities. 5. The t – test value of 2.75 is greater than the critical value at 2.0126 tested at .05 level of significance with 48 degrees of freedom. Hence, the null hypothesis was rejected. This indicates a significant improvement in the respondents’ conceptual understanding through their exposure to the Qualitative Reasoning Approach (QRA). Moreover, using the scores in the problem solving activities at .05 level of significance with 48 degrees of freedom yield a computed value of 5.09. This value is greater than the critical value of t which is 2.0126 that leads to the rejection of the null hypothesis. This means that there exists a Proceedings of the 3rd International Conference of Teaching and Learning (ICTL 2011) INTI International University, Malaysia 8 significant difference in the performance of the groups in the pretest and posttest mean scores in PSA. CONCLUSION Analysis of the data gathered indicated that the QRA in solving worded problems significantly improved not only the students’ conceptual understanding but their problem solving performance as well as revealed by the results of the t – test for two independent samples tested at .05 level of significance. Drawn from this study are the following conclusions; (a) The pretest mean scores of the experimental and control groups are similar and respondents from both groups are low achievers; (b) The post test score of the experimental group is higher than the post test score of the control group in the Problem Solving Test; (c) The pretest mean scores of the experimental group do not significantly differ with the pretest mean scores of the control group. (d) There is a significant difference in the posttest average scores using the Problem Solving Test; and (e) Significant differences exist in the performance of the experimental and control groups on their pretest and posttest mean scores in SCI and in the problem solving test as well. RECOMMENDATION In the light of the findings and conclusions drawn, the following are the recommendations: (a) Encourage teachers to integrate QRA in teaching worded problems not only in Statistics but also in other Mathematics subjects to enhance conceptual understanding; (b) Reinforce the students on the use of written explanation and reasoning as part of their problem solving solution to strengthen their grasp on the concepts; and (c) Conduct further studies using the Qualitative Reasoning Approach (QRA) to further investigate its effect on students’ conceptual understanding and problem solving performance. References Allen, K. (2006). “The Statistics Concept Inventory: Development and Analysis of A Cognitive Assessment Instrument in Statistics”. Dissertation. Oklahoma University. Norman, Oklahoma. Crisostomo, A. (2004). “Students’ Conceptual Understanding and Problem Solving Difficulties in Physics Using Concept-based Problem Solving Strategies”, Master’s Thesis. Technological University of the Philippines, Manila. Lester, F. K., & Kehle, P. E, (2003). “From problem solving to modeling: The evolution of thinking about research on complex mathematical activity”, Mahwah, NJ. Lawrence Erlbaum Associates. Proceedings of the 3rd International Conference of Teaching and Learning (ICTL 2011) INTI International University, Malaysia 9 Lubienski, Sarah T. (2002). “Good Intentions Were Not Enough: Lower SES Students’ Struggles to Learn Mathematics Through Problem Solving. Lessons Learned from Research”, Journal for Research in Mathematics Education. 31. 454 – 482. National Council of Teachers of Mathematics (2000). “Principles and Standards For School Mathematics”. Reston, VA Siguenza, R. (2006). “The Effect of Qualitative Reasoning Approach on Students’ Conceptual Understanding and Problem Solving in Physics”. Master’s Thesis Technological University of the Philippines, Manila. Statistics Education Research Journal Voskoglou, M. (2008). “Problem Solving in Mathematics Education: Recent Trends and Development”, Quaderni di Ricerca in Didattica (Scienze Matematiche) Yeo, K. (2009). “Secondary Students’ Difficulties in Solving Non-Routine Problems”, International Journal for Mathematics Teaching and Learning. astro.temple.edu/~jlynne80/.../BoothKoedingerCogSci2008.pdf core.ecu.edu/psyc/.../docs2210/Research-8-QuasiExpDesign.doc www.childternds.org, 2008