Teaching Geometry Through Problem-Based Learning

New mathematics are discovered and invented everyday and there is a great manifestation of growing recognition of the need among the mathematics educators to increase the emphasis placed on problem solving for all students. (Paja 2001). Mathematics evolved over the past few thousand years in many stages. In high school mathematics in Philippine settings involve elementary algebra, intermediate algebra, geometry and advanced algebra.

All of these were center in answering questions about real life. Ordinary people of all ages are endeavor of mathematics in which they constructs concepts, discover relationships, invent methods, execute algorithms, communicate and solve problems posed by their own real worlds. (Cangelosis, as quoted by Paja 2001). Often times, mathematics is a difficult subject for students who has difficulty in memorizing formulas and using logical thinking.

Students learn hardly anything of what they are taught but if they learn through hands on and minds on approach (Paja, 2001) and if it will incorporate into everyday activities and in subjects like languages arts, science, social studies, visual arts, music, physical education, life skills and performing arts. Posadas, as quoted by Paja 2001 said that they will learn more meaningfully and with longer retention. Students in today’s generation are practical work learners, investigational learners and sociable learner.

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In relevant to that mathematics educators should explore another strategy of teaching that caters the needs of the students that we have today. According to Paja 2001 in his study on Practical Work Strategy in Teaching and Learning Plane Geometry: Its Effects on Students’ Achievement says that utilizing practical work strategy in teaching and learning plane geometry with the aid of manipulative materials has more reaching effect on students’ achievement in mathematical content and process skills than the traditional.

Problem – based learning is a student centered instructional strategy in which student collaboratively solves problems and reflects on their experiences. (http://en. wikipedia. org/wiki/Problem-based_learing). Dewey proposed that education should be built on the child’s interests and experiences (Ernest, 1991). Education becomes meaningful and real to students when it is connected to them personally, as opposed to using materials that may be abstract and unrelated to a child’s lived reality. (Douglas, 1994).

In today’s world of education, lots of researches that conducted promoting and encouraging active learning in mathematics. But still we are searching for other methods in teaching mathematics. In a particular University of Cebu – Main High School Department focuses merely on the tasks of improving students’ competence on conceptual knowledge. In teaching mathematics is usually it is a teacher – centered because less attempts for students to be involve in every class interaction, performing mathematical processes through investigation and discovery thus enhancing mathematics procedural skills.

This study aims to determine the effectiveness of problem – based learning on students’ attitude and performance in high school geometry in which we seek other strategy that caters the kind of learners that we have. It provide teachers with suggested teaching strategies that would greatly influence student’s motivation and enthusiasm as they develop deep understanding of the challenging topics in mathematics. This study also offers a strategy that designed for individual learning and it encouraged to take responsibility of their group and organize and direct learning process with support from an instructor.

Problem – based learning is used to enhance content knowledge and foster the development of communication, problem – solving and self direction learning skills. Problem – based learning (PBL) is an approach to structuring the curriculum which involves confronting students with problems from practice which provide a stimulus for learning. (Buod and Feletti, 1991). They used to enhance content knowledge and foster the development of communication, problem - solving and self directed learning skills.

In PBL classes, students also summarize and present their solutions in a culminating experience. The principle role of the teacher in PBL is that of a facilitator or educational coach guiding the learners in the PBL process. Educator is not the sole resource of information, but instead guides students as they search out appropriate resources. Problem – Based Learning (PBL) is a student – centered instructional strategy in which students collaboratively solve problems and reflect on their experiences, rather than learn primarily through lectures or textbooks.

Problem – based learning require the development of a number of component competences, such as the skills of communication, critical reasoning, logical and analytical approach to problems, reasoned decision making and self – evaluation. (Buod and Feletti, 1991). Engel sees problem – based learning as a means of developing learning for capability rather than learning for the sake of acquiring knowledge. The effectiveness of the PBL depends on the nature of student engagement and the culture of the classroom, as well as the appropriateness of the problem tasks assigned.

Proponents of PBL believe that when students develop their own problem – solving procedures, they are integrating their conceptual knowledge with their procedural skills. (Gilo, 2008). In 1960’s at McMaster Medical School, the PBL approach was started wherein the approach developed by the faculty out of the perceived need to produce graduates who were prepared to deal with the information explosion and who could think critically and solve complex problems. This institution developed its entire curriculum around PBL. (Buod and Feletti, 1991).

However medicine has also been among the pioneers in the application of problem based learning as a means towards rectifying the existing situation in undergraduate medical education (Spaulding as quoted by Buod and Feletti, 1991) and post graduate medical education (Jack and Engel as quoted by Buod and Feletti, 1991). Soon after medicine schools adopted PBL as their center of instruction not later other fields will be using problem – based learning in teaching.

The movement has extended into the K-12 arena as well. Camp, 1996). Educators and administrators of the institution wanted students who could think critically, solve problems and work in teams. And many undergraduate institutions began to develop PBL programs and curricula.

Aalaborg has one of the most comprehensive undergraduate PBL curriculum, and Maastricnt also has a develop PBL program of study. More recently, in the U. S. , the University of Delaware has turned attention toward Problem – based learning, as has Samford University in Birminghan, Alabama.

In addition to these more comprehensive efforts, individual faculty members at more than 300 institutions are using PBL at the undergraduate level (PBL insight, p. 7 as quoted by Gilo, 2008). Through the researches conducted in medical school, we can formulate expectations about the outcomes of problem – based learning. Medical researchers show that problem – based learning provides students with the opportunity to gain theory and content knowledge and comprehension.

According to Schmidt cognitive effects of problem – based learning are the following:

• initial analysis of the problem and activation of prior knowledge through small-group discussion,
• elaboration on prior knowledge and active processing of new information,
• restructuring of knowledge, construction of a semantic network,
• social knowledge construction,
• learning in context
• stimulation of curiosity related to presentation of a relevant problem. Constructivism and Problem – Based Learning

Constructivism is a philosophical view on how we come to understand or know. It is, in our mind, most closely attuned to the pragmatic philosophy of Richard Rorty (1991) as quoted Duffy and Savery, 2001. We will characterize the philosophical view in terms of three primary propositions by Rorty (1991) as well as vonGlaserfeld (1989). Firstly, understanding is in our interactions with the environment. This is the core concept of constructivism. We cannot talk about what is learned separately from how it is learned, as if a variety of experiences all lead to the same understanding.

Learning takes place only through self – activity. (Froebel 1976). Dewey proposed that education should built on the child’s interests and experiences. (Ernest as quoted by Douglas, 1994). Since understanding is an individual construction, we cannot share understandings but rather we can test the degree to which our individual understandings are compatible. An implication of this proposition is that cognition is not just within the individual but rather it is a part of the entire context. (Savery and Duffy, 2001).

Secondly, cognitive conflict or puzzlement is the stimulus for learning and determines the organization and nature of what is learned. When we are in a learning environment, there is some stimulus or goal for learning -- the learner has a purpose for being there. That goal is not only the stimulus for learning, but it is a primary factor in determining what the learner attends to, what prior experience the learner brings to bear in constructing an understanding, and, basically, what understanding is eventually constructed.

In Dewey's terms it is the "problematic" that leads to and is the organizer for learning (Dewey, 1938: Savery and Duffy,2001). For Piaget it is the need for accommodation when current experience cannot be assimilated in existing schema (Piaget, 1977; Savery and Duffy, 2001). Lastly, knowledge evolves through social negotiation and through the evaluation of the viability of individual understandings. The social environment is critical to the development of our individual understanding as well as to the development of the body of propositions we call knowledge.

At the individual level, other individuals are a primary mechanism for testing our understanding. Collaborative groups are important because we can test our own understanding and examine the understanding of others as a mechanism for enriching, interweaving, and expanding our understanding of particular issues or phenomena. As vonGlaserfeld (1989) has noted, other people are the greatest source of alternative views to challenge our current views and hence to serve as the source of puzzlement that stimulates new learning.

In PBL, students learn content, strategies and self – directed learning skills through collaboratively solving problems, reflecting on their experiences, and engaging in self – directed inquiry. It established principles of learning which have been explained through observation and research over the past century, principles such as motivation, relevance, practice, active learning and contextual learning operate significantly in a PBL environment, and to a much lesser extent in conventional curricula.

The effects of problem – based learning on students’ attitude and performance in high school geometry to improve classroom instruction and the quality of education rendered to continuing growing population. This study will benefit the following individual in learning and teaching mathematics in different approach. Students. The students were the primary reason of this study. They would be directly affected with the benefits of the study since they were the focus of it.

Students would be relieved of the conventional classroom structure which they perceive as boring and unmotivating. They are given highly appropriate learning experiences to build positive attitude and productive individual. Teachers. The findings of this study would be of great help to the teachers not only in mathematics but also in some related areas of concern. Teachers at all levels would be able to select appropriate teaching techniques that complement problem – based learning.

School administrators would likewise be benefited by the outcomes of the study and would capture an insight and opportunity to include in the present scope of the mathematics program and help improve the curricular content to adopt students’ level of learning with the same weight, being in the position, they have the chance to persuade the teachers to adopt problem – based learning as part of students’ learning experiences.

The result of this study would also enable the curriculum writers to redesign or restructure curriculum materials which could better facilitate mathematics learning through problem – based earning. Scope and Delimitation Content Delimitation The area of the study was the level of performance in high school geometry and the attitudes towards problem – based learning of the third year high school students of University of Cebu High School Department – Main Campus in the school year 2010 – 2011. Place Delimitation This study was limited to University of Cebu High School Department – Main Campus located in corner Sanciangko and Juan Luna Streets, Cebu City. Time Delimitation The study conducted in the school year 2010 – 2011.

A teaching working from a multiculture, social – reconstructivist approach attempts to create a learning environment that is as democratic and open as the power asymmetries of the classroom allow, but with explicit recognition of this asymmetry. (Ernest 1991 as quoted by Douglas, 1994). Mathematics is our general education component that entails enriching a personal knowledge of the students that includes the opportunity to develop the power to explore, make conjecture and reason logically.

This component helps students to become broadly educated, creative, cultured, morally pright and productive citizens. (Paja, 2001). Mathematics as an interdisciplinary language and tool. Mathematics can be used to help represent, communicate about, and solve problems in many different disciplines. Many jobs and other aspects of responsible adult life in our society require some mathematical knowledge and skills.

According to Stephien and Rosenthal (1992) that PBL instruction is designed to provide students with a guided experiences in solving an ill – structured problem. It orienting students toward meaning – making over fact – collecting. They learn via contextualized problem sets and situation. (Rhem, 1998). Ulmer says, this approach gives students immediate feedback. “It keeps a constant flow going between teacher and student, and you cant’t put a price tag on that. According to the study of Gilo (2008), that PBL can produce socially responsible citizens. This gives the youth a sense of awareness and participation in the community.

They love the challenge being the problem solvers and it gives a sense of accomplishment for having been part of the society they belong. PBL is a motivating way to learn as learners are involved in active learning, working with real problems and what they have to learn in their study is seen as important and relevant to their own lives.

According to Spence that problem based learning gives you opportunities to examine and try out what you already know; discover what you need to learn; develop your people skills for achieving higher performance in teams; improve your writing and speaking abilities, to state and defend with sound arguments and evidence your own ideas; and to become more flexible in your approach to problems that surprise and dismay others. Despite the work and effort it requires, PBL is never dull and is often fun.

Problem – Based Learning proponents emphasize that it improves thinking and learning skills and cognitive abilities in students. It has been reported that PBL – trained students are more frequent users of libraries and other information resources, which support independent learning. They acquire life long study skills, especially in their early years of study, giving rise to sustained learning. PBL educated students have a more holistic approach to their subject, more readily integrate new information, adapt to change and work well as member of a team.

Generally PBL appears to increase students interest and enjoyment to the subject and enhance their professional development. (Gilo, 2008). Chapter 3 Research Methodology Methodology This study utilized the quasi – experimental method with a content group and an experimental group using the pre – post tests. The quasi – experimental method was used since the subjects in each group were matched in terms of some selected variables such as classroom setting, classroom environment, instruction and academic performance. The study conducted on the second quarter grade.

The two groups were given the pre – test on solid figures and its measurements to determine the mathematical achievement of each student. Intervention took place after conducting the pre – test which lasted for two weeks. The traditional way of teaching was given to the control group and the problem – based learning for the experimental group. After four weeks of experimentation a post – test was administered to determine the changes in their performance in high school geometry specifically in solid figures and its measurements.

Research Environment This research was conducted in University of Cebu – High School Department Main Campus is located at the corner of Sanciangko and Juan Luna Streets, Cebu City. It is private non – sectarian institution of higher learning. It provides the learners with the essential knowledge, skills and attitudes that allow them to improve their quality of life and increase their opportunities to participate in and benefit from social and economic development.

It aspires to provide the learners with academic, scientific, technical and vocational, knowledge, skills and attitudes essential in meeting the demands of time, enhance the individuals emotional, social – cultural and spiritual needs; deeper the learners’ awareness and willingness to be pro – active in community projects and activities including environment protection and preservation, produce graduates who are highly qualified for the world of work. And as testament of its desire to provide est education for the masses, it is now the fastest growing university, if not the most dynamic among all the universities in the city of Cebu in terms of the number of students coming from public and private schools in urban or rural areas.

The instruments in this study were the Mathematics Performance Test and an adopted Mathematics Attitude Scale. The achievement test was a teacher – made test about the high school geometry particularly solid figures and its measurement. To measure students’ attitude in mathematics, Mathematics Attitude Scale was used. This mathematics attitude scale was adopted from the study of Ruyca, 1994. It consisted of 20 positive and negative statements. This attitude questionnaire is a 5 – point Likert scale to which the subjects indicate SA for strongly agree, A for agree, U for undecided, D for disagree and SD for strongly disagree.

For reliability of the said test, split – half method was used. A coefficient of correlation of 0. 74 described that the test was highly reliable. Research Procedures The procedures of the study were done through data gathering and treatment of data. Data Gathering The researcher secured a written permit with the approval of the high school principal of University of Cebu –Main Campus. After the written permits signed and approved by the authorities, the researcher administered a 40 – item teacher – made test to the 40 third year high school students of University of Cebu – Main who are not respondents of the study to establish the validity of the test.

When the test was found to be reliable and valid, a pre – test was administered to the actual respondents, the third years Sapphire (control group and Jade (experimental group), a week before the experimentation started.

All the discussions and presentations were delivered by the teacher. The pacing of the lesson depended on the teacher’s evaluation of the students’ performance in their previous activity. Each lesson was taught for not more than two meetings. After a month of experimentation, a post – test was given to evaluate whether the students in the control group gained knowledge in solving the areas, surface area and volume of a plane and solid figures. On the other hand, the 41 III – Jade students were exposed to problem – based learning. In the PBL, the learner will be given a problem and they were attempting to answer it of the information of what they already know.

They will identify what they need to learn to better understand the problem an how to resolve it. Once they have worked with the problem and identified what they need to learn, the learners engage in self-directed study to research the information needed by finding and using a variety of information resources (books, journals, reports, online information, and a variety of people with appropriate areas of expertise).

The learners then return to the problem and apply what they learned to their work with the problem in order to more fully understand and resolve the problem. After they have finished their problem work the learners assess themselves and each other to develop skills in self-assessment and the constructive assessment of peers.

Self-assessment is a skill essential to effective independent learning. The faculty in turn become resources, tutors, and evaluators, guiding the students in their problem solving efforts. To measure the attitude of each student towards mathematics who had some through problem – based learning and traditional method, a Mathematics Attitude Scale the level of interest, feeling, perception and trend of attitude of the student towards high school geometry in both control and experimental groups were gathered.

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