Views of Van Hiele towards Teaching and Learning Mathematics - Essay Example

Views of van Hiele towards Teaching and Learning Mathematics Name: Course: Tutor: Date: Abstract Through a critical analysis of van Hiele’s works, the paper explains how teachers or instructors can teach geometry in both primary and secondary schools. The paper reports on the suitability of teaching phases depicted in van Hiele model within the classroom context. The paper focuses on van Hiele theory which underscores levels of thinking in an attempt to understanding geometry. It gives precise information about the background of Heile’s work and the relationship between the five levels of thinking. Introduction In the recent curriculum reforms, teaching and learning geometry has been emphasized in many parts of the world. The basic geometry taught at primary schools are useful because they enable students acquire relevant experience and skills applicable in axiom-theorem mathematics at later stages of learning i.e. at secondary level. In developed countries, the method of geometry proofs has been embraced as an integral part of curriculum in the learning institutions. Many students have encountered numerous problems in understanding geometry in mathematics, and this is closely associated with the lack of fundamental concepts. Broadly speaking, teachers or instructors are concern with imparting correct information about particular subjects to the learners. As asserted by scholars and educationists, teaching has direct influence on the learning process of an individual. In the course of teaching, instructors interfere with the learning process of the subjects. In order to pursue intelligent teaching interventions, the concern authorities must possess sufficient knowledge and skills and the theoretical aspects that affect the process of human learning. Throughout his career as a teacher, van Hiele observes that students encounter great deal of difficulties when undertaking geometrical mathematics. From his observations, it is explicit that geometry learned in secondary schools is more complex than primary school geometry thus calls for higher level of thinking. Consequently, this sparked loads of debates and discussions on the abilities of students to learn geometry. From intensive discussions, it became apparent to Heile that students progress in sequence of levels during the learning process, particular in geometry. According to the theorist, the progress ranges from analytical thinking to rigorous deductions. With this realization, he concluded that an individual can only transit to another level of learning after passing through rigorous learning process in the previous levels (Van Hiele, 1986). In an attempt to support this approach of levels and phases of learning, Heile defined some important aspects that affect the process of learning among the students. At the same time, he supported that teacher’s duty is to help student pass through various levels of thinking involved in understanding the subject of concern. Hiele worked toward understanding various levels of thinking when learning geometry. He further argues that Hiele focused on the roles of instructions in guiding the students pass through various levels of thinking As the basis of his works, Heile suggested that a student cannot attain a level of thinking only by mere thinking but by incorporating appropriate alternatives of exercises that ought to be created by the instructing authorities in the learning environment (Van Hiele, 1986). After series of geometry works and subsequent observations, Hiele (1986) criticized the available techniques of teaching arguing that they are ineffective in dissemination of geometry facts and information hence compromising clear understanding amongst the learners. He argues that teachers should focus on inculcating insights among the students as a way of helping them proceed from one level to the next. To promote better understanding of geometry in mathematics, students must develop insights that aid in learning structures. He disparages past methods of learning geometry on the basis that they entailed more of facts than structures. He determined that the learning mathematics in various levels of school involves more than mere learning of facts. Therefore, development of insights helps attain the real purpose of teaching i.e. learning of structures. Arguably, learners particularly in secondary schools conceptualize structures upon acquiring relevant insights. According to Hiele (1986), a structure enables human beings act decisively in situations that they have never encountered before. It is vital phenomenon because it helps them pursue their objectives persistently. Through structures, people understand and relate with each other and the environment within which they live. By adopting continuing structure in a similar pattern in the learning environment, the stakeholders in the learning environment can easily express their joy and harmony. Hiele’s theory reveals that the strength of structures are measured basing on the element of rigidity, and this affects the probability at which they can be continued Extending feeble structures is often accompanied by great deal of mistakes. Notably, Hiele treated mathematical structures very rigid. This implies that they are very strong when the rule of structure is applied (Van Hiele, 1986). In the quest of developing his ideas of mathematical structure, Hiele adopted Gestalt psychology which supports that mathematical structures consisted of important features. In reference to the Gestalt contributions, Hiele believes that a structure can be extended, however, the degree of extension can only be ascertain by the learners who have full knowledge on components and features of structure. Gestalt psychology agitates that a structure can be viewed as finer components. However, its determination is not influenced by the rules of the game provided that its originality is adhered to. This allows more details to be incorporated when building up a particular mathematical structure. While following the Gestalt psychology, Hiele (1986) maintains that strong structures are more original than feeble structures. The original structure of a more inclusive structure has specific rules which delineate its originality. Gestalt psychology further proposes that a structure can be isomorphic with a particular structure thus using rules that correspond to each other (Van Hiele, 1986). To clarify Gestalt psychology propositions, Hiele (1986) used human skeleton as basis of reference. He stated that by studying human skeleton “The extension of the structure may happen when we realize that we have such skeleton ourselves (p.29). Therefore, finer structure is developed when man gives names to certain parts of the human skeleton. Van Hiele suggested that properties of a structure can either inmate in human beings or can be discovered after undertaking particular study. Therefore, pupils and students should be stimulated to develop their own recognition of some properties of phenomenon structure in order to enhance positive development of insights. Furthermore, perceptive insights and understanding can be achieved if students are well versed with the nature of the structures (Van Hiele, 1986). The process of learning can be thorough only if the individuals grasp the fundamentals of the subject matter. In the real scenario, students can explain what they have learned without any fear if they are well versed with the core areas of the subject matter. Conspicuously, many students around the world have run into great deal of problems understanding geometry due to lack of simple set of axioms and principles. In order to foster ease understanding of the geometrical mathematics, teachers or instructors ought to clarify the axioms or the properties of the structures used in geometry. Students’ inability to comprehend the specific features of the structures would hamper the process of learning the subject. By observing the properties of a structure, students will automatically know how to apply the prescribed set of principles in arriving at desired outcomes (Van Hiele, 1986). The van Hiele Levels According to van Hiele (1986), a student passes through five levels of thinking in the course of learning. These phases are chronological and hierarchical. Basing on the factor of originality, van Hiele labels the thinking process using numbers 1 to 5, and this has been used up to date. Van Hiele uses geometrical examples to show the various levels of thinking. Using his past works, Hiele comments on the complications encountered when tracing the levels of thinking. This is attributed to the fact that these levels are based on the thoughts of human beings Human thinking complexion can only be understood by considering the disagreements as to what the real number of the levels of thinking should be. As such, teachers ought to know how the learners think. In essence, the functioning of students’ brain affects their ability to learn and understand the subject of discussion. Understanding of geometry figures calls for adoption of element of appearance. This helps them to identify, name and make explicit comparison and distinction between various geometrical figures as the basis of understanding mathematics. According to Hiele (1986), students who can only recognize specific shapes of figures without paying minimal attention to important components are reasoning at lower level of thinking (level 1). For instance, a student may recognize square since it resembles a window and not because it has four equal straight sides. In such context, the student does not appreciate the real components of a square. In the second level of thinking, students are capable of discovering properties of specific shapes in a more holistic way. This may involve analysis of figures while considering basic components used to best describe certain figures. At this level, students are assumed to be reasoning meticulously hence they are capable of giving actual attributes of particular figures of geometry. Significantly, geometric properties are the object of thought which have been learnt at earlier stages. Van Hiele recommends that should students understand the relationship between the objects and set of properties in order to satisfactorily pursue geometrical mathematics. They should appreciate the fact that set of properties of one object may imply properties of another object. The level calls for students’ abilities to deal with simple arguments regarding simple geometric figures. They must be proficient when describing particular geometric figures. They may argue that isosceles triangles are symmetry thus have similar base angles. However, this specific student may not be able to understand that a figure belongs to several classes i.e. a square is oblong. Therefore, a student at this level is capable of stating a definition without understanding of the actual features of the subject matter under discussion (Van Hiele, 1986). In the analysis level, the development of network of interrelated concepts, ideas and properties are accomplished and stabilized at level three (ordering). At this stage, students are capable of giving precise definitions and informal arguments about certain concepts and properties of geometrical figures. They understand the relationships between figures and can use these deductions to make explicit observations made at analysis level. Despite clear-cut understanding of the quintessence of geometry, the student at this level is incapable of giving proofs why certain figure appears the way they are. In a classroom setting, teachers and other instructing authorities ought to ensure students are competent to carry out deductive reasoning. This allows them to learn how to combine ideas and thereby providing substantial proofs of the associations. In secondary schools, the student should be in a position to easily construct geometric proofs by following teacher’s instructions. Students in high level of thinking must defined terms, axioms and theorems of geometry. They ought to define these terms basing on their understanding. Note mentioning, students at this level argue that certain definitions and axioms are not arbitrary hence cannot be clearly comprehended. This level is characterized by students’ abilities to develop clear definitions of figures. The students can easily study geometry as an integral component of mathematical system as opposed to mere understanding of shapes (Van Hiele, 1986). Hiele points out that student can establish theorems in various mathematical systems and compares them when at level 5. At this stage, students have clear-cut understanding of abstract geometry exclusive of pictorial models. Significantly, the axioms (postulates) are subjected to critical analysis as part of abstraction at this level (Van Hiele, 1986). The five levels of learning are substantial during provision of framework for teaching or giving out instructions directed towards development of clear understanding of the material under discussion. Van Hiele asserts that a student can only pass through the five levels of thinking under the presence of health assistance from instructors (Van Hiele, 1986). The van Hiele theory has drawn the attention of many scholars and educationists around the world. Some have sought to know whether and individual can skip certain levels of thinking. Van Hiele (1986) theorizes that a learner cannot attain one level of thinking without visibly understanding all the previous levels. This view has been supported by research in various parts of the world. According Hiele, there is one exception for this rule. Some talented fast learners appear to skip levels because of their abilities to develop logical reasoning skills without relying on the basic concepts of geometry. In case of rote learning, instructors are obliged to supply new information so as to assist the student operating in levels which they do not belong. Teachers with limited knowledge on geometry commonly adopt level-reduction method. However, this has direct negative effects on the learners of geometry. In order for real geometry to be taught in classroom, teachers ought to understand and recognize the level at which the students are operating (Van Hiele, 1986). There are circumstances when the instructor is thinking at a different level of thinking than the students. This situation is common high school where geometry teachers think at higher level whereas new students think at first level. Imperatively, teachers ought to understand that their interpretation of certain geometrical aspects is quite different from that of students despite using similar wordings. Teachers should be keen to ensure that they understand students’ level of thinking before interpreting certain concepts or topics as this will facilitate effective communication (van Hiele, 1986). The van Hiele (1986) determined the features of levels of thinking which were later modified by Usiskin as adjacency, distinction and separation. Inherently, Hiele theory suggests that an individual can only comprehend geometry after passing through all the levels of thinking, and this should occur in a specific order. Currently, there are great interests in categorization of different levels of thinking especially in the learning environment. However, this has been marred by divergent views and lack of consensus of opinion on whether or not the five levels of thinking exist and are hierarchical in nature. Noticeably, the ongoing research has widely adopted the five levels of thinking as postulated in Hiele’s theory. In summary, the Hiele’s levels of thinking are sequential, every level comprises of certain set of symbols, the passage through the levels depends entirely on the instructional experience, students in inappropriate level are subjected new information and that subject inherent at one level is unambiguous in another one. Van Hiele pointed out that an individual has to go through various phases in order to achieve the highest level of thinking (van Hiele, 1986). How can a teacher assess the student’s level of thinking? For effective learning process to be achieved, teachers are obliged to understand the van Hiele level of students. Teachers can design tests which are then assigned to various van Hiele levels. In a classroom environment, teachers can assess the student’s level by analyzing their responses to certain geometric tasks. For instance, an instructor can keenly observe how an individual learner uses particular geometric language. Lesson analysis using teaching phases model According to Hiele (1986), students must pass through five phases as part of learning processes before proceeding to another level of thinking. Students progress from one level to another after accomplishing the five phases namely information, guided orientation, explicitation, free orientation and integration (Van Hiele, 1986). The information phase entails students getting acquainted with the subject matter. In a primary school setting, the teacher should understand that learners are at lower level of thinking. Arithmetic lesson should be introduced with great deal of care. As part of brief introduction to the lesson, the teacher should use simple concepts of mathematics to facilitate effective learning process. The teacher should avoid complex examples that might sound thorny to the learners. Guided orientation is the second phase and entails students engaging in exploration of particular objects in well structured tasks such as constructing, observation of asymmetry, measuring and folding. In a primary school setting, the instructor should give the pupils simple tasks to perform. The pupils should utilize the fundamentals of mathematics i.e. addition and subtraction (Van Hiele, 1986). Explicitation phase demands that student give best description of what they have learned concerning specific topic using their own words. They can express them in required words and using technical language of the subject matter. In the primary setting, it is important to teach pupils how to express figures or solutions in words. The pupil should be able to differentiate between multiplication and addition through expression of words. It is at this phase that the teacher should introduce relevant but limited mathematical concepts. They should be able to lead the children in discussions so as to enable them understand how to use the languages appropriately. At this phase, they should be in a position to express ideas concerning properties of figures at hand. After explicitation, students pass through free orientation phase where they learn by engaging themselves in more complex activities. They also identify properties of shapes or figures thereby enabling them investigate their properties. At integration phase, the students should be able to summarize what has been taught about the subject. This will enable them reflect on their actions thereby obtaining an overview of newly established network of relations. The students should be allowed to reflect on their actions by memorizing them. Upon completion of the five phases, the student should be able to memorize some vital characteristics of the subject learnt. This is preceded by learning process. At this point, the instructor should create a conducive environment that enables learners achieve the higher level of thinking. Great care should be taken by the instructors in order to avoid phasing out of students. An effective teacher should be able to identify the phase of learning that a child is in. A child can understand his/her abilities if he or she engages in the appropriate phase (Van Hiele, 1986). Effective teachers must understand the nature of each phase in order to effectively delivery their services. In the first phase, teachers ought to place learning materials at the disposal of the learners. The materials should explicitly clarify the subject matter of discussion. The second phase demands the teacher to avail materials to be used by students in understanding the relationship between particular properties of the field of thinking. The implications of van Hiele theory for teachers’ practices According to van Hiele (1986), effective learning process occurs only when the learners appreciates the objects of study in pertinent contexts. This can only be achieved through active engagement in class discussion and reflection. In the realm world, learning has not been effective because most teachers use lecture and memorization as the valuable techniques of teaching. Competent teachers expose student to relevant experiences and opportunities for intensive discussion. An effectual teacher assesses students’ levels of thinking before providing instruction. These instructions should be provided while considering phases of learning in order to aid in development of successive level of unequivocal understanding. At intermediate levels, teachers should employ appropriate applications method in order to allow student capture valuable properties of geometrical figures (Van Hiele, 1986). Conclusion As a general theory of mathematics, van Hiele’s theory has been widely used in explicating the various levels of thinking that are closely associated geometry analysis. The theory seeks to provide explicit understanding of the movement between ‘levels of thinking’ as major concerns of many teachers around the world. Reference Van Hiele, P. (1986). Structure and Insight: A Theory of Mathematics Education. USA: Academic Press Publishers. Read More