Effects of experiential learning approach on mathematical creativity among secondary school students

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1075.2017-0124 Educational Sci., 2017, Vol. 62, Iss. 6, pp. 19-27 This paper is available online at EFFECTS OF EXPERIENTIAL LEARNING APPROACH ONMATHEMATICAL CREATIVITY AMONG SECONDARY SCHOOL STUDENTS Nguyen Huu Tuyen Bac Ninh Pedagogical College Abstract. Creativity is human copious activity. It is one of the goals of teaching mathematics at secondary school. At the age of secondary students, the way of thinking has changed into an intellectual, logical and formal one. Mathematics is a subject which seeks the understanding of all patterns permeating both the world around us and the mind within us. Experiential LearningApproach (ELA) onMathematical Creativity at secondary school offers students a lot of opportunities to experience creativity. This article goes further into the effects and the positive impacts on Mathematical Creativity of ELA among Secondary School Students. Keywords: Experiential Learning Approach, Mathematical Creativity, Experiential transformation, Experiential learning model, Teaching Mathematics at secondary school 1. Introduction Experiential Learning Approach (ELA) originated from ancient times with the typical example: the Confucius educational phylosophy (551-479 BC) with the conception "What I do, I will understand", Xocrat (470-399) paid a special attention to learning by doing specific work. When Vygotsky (1896-1934) - a Russian psychologist - presented the idea of "the nearest developmental area" where it contained personal experience, and served as a basis for learners thanks to previous learning and experience. In the 19th century, spychologists and educators in the world such as John Dewey, Kurt Lewin, Jean Piaget, lev Vygotsky, David Kolb, William James, Carl Jung, Paulo Freire, Carl Rogers. . . [7] deeply and systematically carried out the researches on ELA in some aspects. Experiential learning theory, officially made public for the first time by David Kolb in 1971, was quite comprehensive as a learning approach based on experience, accumulation of experience and experiential transformation. It has been applied in more than thirty fields and disciplines since then. Its principles, definition and process have been widely used to develop and popularize the curriculum of secondary schools, university education and professional training [8]. Nowadays, experiential education is one of the tendencies of advanced educations. According to the draft curriculum for secondary education, experiential activity is not only a subject - one of three main elements of the new secondary curriculum- but also an approach to innovate teaching methods in all of the subjects to achieve educational objectives. Mathematics at secondary schools has a lot of opportunities to apply Kolb’s theory of Experiential learning approach. Hence it brings about creative thinking to students - the core activity of teaching Received date: 25/3/2017. Published date: 10/6/2017. Contact: Nguyen Huu Tuyen, e-mail: nguyenhuutuyen.bacninh@moet.edu.vn 19 Nguyen Huu Tuyen mathematics at secondary school. Experiential learning is similar to "learning through doing", but the conception of this theory states that learning is a process creating new knowledge based on real experience (an interactive environment, real life models, knowledge, skills, approaches, done mathematical tasks, surrounding people’s experience) (Concrete experience). Basing on the evaluation and analysis of the available knowledge and experience, students first observe, analyse and reflect (Reflective observation). Then students feel their ways, predict, choose the suitable solutions to given tasks and find out new knowledge (Conceptualization). Finally, students actively apply in new situations (Active Experimentation). Experiential learning is considered to be contrary to academic learning - a process to get knowledge through studying a matter without direct experience. Up to present, the application of experiential learning approach in some countries from all over the world as well as the announcement of the domestic and overseas researches on this field have only focused on extra -lesson activities in the society and out of the mathematical content in curriculum. This research states the conception of ELA in teaching mathematics at secondary school. It also analyses and makes mathematical creativity and the effects of ELA on mathematical creativity clear. Moreover, this reasearch analyses the mentioned content above by an illustruting example, which is the most difficult and desired problem now because it has not been clearly mentioned and there has not been any document published. On the basis of the example, teachers can examine to design a lot of lessons in the mathematics’ curriculum of secondary education. 2. Content 2.1. Mathematical creativity According to Mann, creativity has been proposed as one of the major components to be included in the 21st century education [3]. There are various ways to define mathematical creativity. Among them, there is a commonly agreed definition that mathematical creativity is a novel way of thinking characterised by fluency, flexibility, originality and elaboration [4-7]. According to this, fluency is the number of responses in which a learner can give to a mathematical question, the number of related ideas. It shows the ability to give several different responses to a mathematical task that relates to the coherence of the ideas, flow of association and use of basic and universal knowledge [3, 5]. Flexibility is the shift in the caterories in the responses to a given mathematical task; it is the number of catergories or classes in a learner’s pool of ideas and responses. In other word, it may be defined as the ability to generate a wide range of ideas and a variety of solutions [4, 5]. Originality is defined as statistical infrequency. It is characterised by a unique way of thinking and unique products of mental activity [5]. This is when responses are novel compared to others to the same mathematical task. Elaboration is the ability of a learner to produce detailed steps [8]. It is building on other ideas. It refers to the number of details in solving a problem [3]. Mathematical creativity is an essential aspect in the formation and the development of mathematical talent [3] as well as constructing mathematical knowledge. Mathematical creativity is a construct involving divergent and convergent thinking, problem finding and problem solving, self expression, intrinsic motivation, a questioning attitude and self confidence [9]. Therefore, the main goal of mathematics education is the "mathematisation" of the children’s thinking. Clarity of thought and pursuing assumptions to logical conclusions is the central to mathematical enterprise introduced a criterion for measuring mathematical creative ability [9, 10]. He addressed both convergent thinking characterised by determining patterns and breaking from established mind set and divergent thinking defined as formulating mathematical hypotheses, evaluating unusal mathematical ideas, sensing what is missing from a problem and splitting general problems into specific sub problems. 20 Effects of experiential learning approach on mathematical creativity among... 2.2. Experiential learning in teaching mathematics "Experiential education" has been introduced to the modern educations in a lot of countries in the world since the early 20th century. In 1977, with the foundation of Association for Experiential Education (AEE), Experiential education was officially recognised and publicly stated. At the summit conference of the United Nations about "2002 stable development" UNESCO approved the program named "Teaching for a stable future". In that, an important part of "Experiential education" was introduced, popularised and widely developed. Nowadays, UNESCO acknowledges "Experiential education" as a bright prospect in coming decades. The opinion of learning through experiencing has become an education mainstream way of thinking associated with the psychologists and educators such as John Dewey, Kurt Lewin, Jean Piaget, lev Vygotsky, David Kolb, William James, Carl Jung, Paulo Freire, Carl Rogers. . . These days the idea "Learning through doing, learning through experiencing" was still an American typical education philosophy. According to International Association for Experiential Education "Experiential education is a catergory consisting of a variety of approaches. There, teachers encourage learners to experience, then reflect, summarize to reinforce their knowledge, develop skills, form living values and develop their own potential, contribute to community and society actively". Educators here can be teachers, volunteers, guiders, trainers or psychological doctors etc. This shows the simplicity, the diversity, the popularization and the application of "Experiential education”. In my opinion, Experiential learning in teaching mathematics at secondary school is a process that students themselves directly feel their ways, predict and find out new mathematical knowledge basing on their own experience. Students gradually transform their learning experience to widen their knowledge, to broaden the value system and to change their lifestyle. Experiential learning approach asserts that acquisition of skills and constructions of knowledge by the learners is direct result of experience. The learner is said to have the ability to select and participate in experiences that will further their growth [11]. Experiential learning can exist without a teacher and relates solely to the meaning making progress of the individuals’ direct experience. This is in agreement with Roger’s opinion (1969) [12], he asserts that experiential learning is equipvalent to personal growth and change. According to Newsome, Wardlow and Johnson (2005) experiential learning approach elevates students’ recognition levels, increases use of critical thinking skills and therefore enhances students’ ability to obtain, retain and retrieve knowledge hence increased achievement. Learning is a cycle that begins with experience continues with reflection and later leads to action which itself becomes a concrete experience for reflections. David Kolb (1984) developed a learning model based on experiential learning which is often known as the Kolb’s learning model [2]. It is the inheritance and the suppliment of Lewin’s model of action research and laboratory training, of Deway’s learning model and Piaget’s learning and concious development model. Kolb’s opinion is consistent with the models above. It closely connects with the intellectual initiation in Deway, Lewin and Piaget’s works. Moreover it puts the emphasis on the central role of experience in learning process. It aims at "processing learning" with clearly defined stages and actions. Through this process of learning, both teachers and learners can continuously improve the learning levels. This is one of the models that are widely used in envisaging curriculums, planning lessons, in training and instructing for tertiary education courses. In this process, Kolb recommended that the order of experiential learning model is followed, but it is not necessary to start at the certain stage of the process. However, Kolb based on an important assumption of learning - knowledge originates from experience. Knowledge needs creating (or re-creating) by learners not remembering what has existed. Therefore, Kolb’s process should be used correctly to gain the greatest effect. Kolb and other researchers further realized that the choice of the point to start and the bias 21 Nguyen Huu Tuyen The Kolb’s experiential learning model in favor a certain stage reflect the learning method of each learner (or each subject). The basic point of view of this learning model based on experience is that learners need to reflect their own experience then generalize and formularize the ideas so that they can apply these ideas in real world to see whether they are right or wrong, useful or useless etc... After that, the learners have got new experiences to get started for the next learning process. The processes are repeated until the planned objectives can be achieved. This process requires learners have discipline in learning by planning, doing, reflecting and applying theory. Detail description of the stages of Kolb’s process. Concrete Experience: Learners can have some experience by reading material, attending lectures, watching videos relating to the topic that they are learning or trying following the instructions of some introductory lessons, or self - performing with the teachers’ assistance. All of those factors help students have a certain experience that then becomes an important input material of the learning process. This stage provides the basis for the learning process in which the lessons engage the individual personally; learning relies on open mindedness and adaptability rather than a systematic approach to stituation and problem. There is involvement in personal experiences and an emphasis on feeling over thinking. Creative work involves a certain amount of pre-existing domain knowledge and its transformation into new knowledge [13]. The role of the teachers is describing the activity and learners do it. However, the most important experience is the one that learners can feel on their own. Reflective observation: Learners need to analyse and evaluate the existing facts and experience. There must be reflection in this stage that is learners self- consider the experience carefully to see how they feel, whether they can understand the ideas or not, whether it is logical or not, whether they choose the right way or not etc... In learning process, reflection deeply implies that learners always ask themselves and give answers to a question: "Is this way going on well?" and purely use their intuition to answer the question. In the reflection process, learners take down the consideration in a natural way on their own. Hence, learners not only draw 22 Effects of experiential learning approach on mathematical creativity among... themselves some lessons but also have new orientations to make the next stage more interesting and effective. For teaching, teachers use the same technique for both teaching and learning so as to gain effective sollutions and activity. There are some kinds of reflection that apply more deeply than consultation, analysis or generalization from different sources and bring forward the evaluation of the experience. When reflecting, learners actively take part in the learning process, therefore learning is helped. If reflection is good enough, it will help learners improve, enhance and regulate the learning growth. Learners develop logical thoughts, verbalize those thoughts relate to others in the group and compare experiences and opinions. The applications of classroom knowledge in the context of real life situations are the focus of learning [3]. The role of the teacher is to promote the atmosphere of acceptance of individual participants and diverse thinking, to design activities that help learners to construct meaning and take the initiative becoming more creative in mathematical learning. Conceptualization: After experiencing detail observation and deep consideration, learners conceptualize their gained experience. New ideas are derived from experiences. This stage is very important for the experience to transform into knowledge and a set of ideas kept in the brains. In this stage, learners assimilate and distil the observation and refections into a theory. The students come to understand the general concept of which their concrete experience was one example by assembling their experience into a general model. Abstract conceptualization requires student to use logic and a systematic approach to problem solving. There is an emphasis on thinking manipualiton of abstract symbols and tendency to neat and precise conceptual systems. The students share their reactions and observations about their experiences. The learners at this stage provide answers to the question arising from the experiences by providing solutions and making generalizations. According to National Council of Mathematics Teachers (2000), the abilities to solve a problem with several strategies or the abilities to reach different answers in a specific task are valuable evidences of the development of mathematical reasoning. Without this stage, the experiences can not be improved and developed into a more helpful new level, but it is only concrete experience gained during the learning and practicing process. Abstract conceptualization ends with our making plan for the next activity in the coming time. This stage usually proceeds the last stage naturally (reflective observation) by answering the important questions arising during reflective observation stage. It can be considered as the conclusion of the previous stage and the next one will be the stage to verify its accuracy. Active Experimentation: In the last stage, the learners came to a conclusion based on reality with closely associated basis and thoughts. This conclusion can be regarded as a theory and we have to apply it in the real world to test. This is very vital for the constitution of real knowledge. According to Kolb and some constructivists, universal truth needs comprehensing or verifying. This is the last stage for us to confirm or disclaim the concepts of the previous stages. In this stage, students use the theories they developed during the abstract conceptualization stage to make predictions about the real world situations. They connect subject matter and life skills discussion to the larger world. Students’ action and wishes are new concrete experiences. The learners are expected to use or test the conclusion, generalizations and solutions in new situations (Kolb&Kolb, 2008). The learner involvement facilitates personal growth and skill development, giving a measure of empowerment to the learners [2]. 2.3. Effects of experiential learning approach on mathematical creativity among secondary students The use of Experiential learning approach on teaching mathematics among secondary students gives learners an opportunity to become more creative in mathematics by constructing meaning and having a critical mathematical thinking. Experiential learning approach also helps 23 Nguyen Huu Tuyen students to develop abilities to solve problems with several strategies or the ability to reach different answers in a specific task which are valuable evidences of the development of mathematical reasoning. Mathematical Creativity at classroom setting is the process that results in novel and insightful solutions to a given problem and the formulation of new questions and possibilities that allow an old problem to be regarded from a new point of view [3, 14]. There is a need to come up with the teaching methods that will enhance Mathematical Creativity. It is experiential learning. Experiential learning offers a critical link between the classroom and the real world. The findings of this study are in agreement with those of Casanovas, Miralles, Gomez and Garcia [3, 15] who noted that science learning based on the experiential learning model promotes students’ instruction of scientific knowledge and increase the fluency and flexibility of ideas generated. Mathematical Creativity is the ability to solve problems or to develop thinking structures, taking into account the peculiar logical deductive nature of the discipline and of the fitness of the generated concepts to integrate into the core of what is important in Mathematics. According to Stoyanova and Ellerton creative thinking ability and exp