Justification of Process Oriented Methods in the Teaching of Fractions - Essay Example

ASSIGNMENT TWO: DESCRIPTION AND JUSTIFICATION OF PROCESS ORIENTED METHODS IN THE TEACHING OF FRACTIONS 1. Introduction Fractions are required for geometry and numbers, which makes up the essence of Mathematics. However, many students continue to experience difficulties when it comes to understanding them, hence the existence of a notion that fractions are ‘hard’. This paper mentions some of the conceptual difficulties that students have when dealing with fractions and some of the ways that they can be overcome. It also describes some of the methods that exist that are used to teach fractions in schools. Furthermore, this paper continues to offer one approach that is considered highly effective in the teaching of fractions and provides justification for it. Additionally, some teaching strategies will be mentioned. 2. Overview of students’ conceptual complexities on fractions In order for students to completely understand fractions and their operations, they must first understand fractions and their equivalents. They also need to be capable of carrying out the four operations of whole numbers, which include addition, subtraction, multiplication and division. However, this understanding has been considerably affected by negative attitudes towards fractions. These attitudes are the result of the students not being able to fully comprehend the concept of fractions. The main conceptual complexity of students on fractions is that students fail to understand the different ideas that are involved in rational and natural numbers. This inability is what makes them think that fractions are ‘difficult’. For example, students may think that ¼ is larger than ½ simply because 4 is greater than 2.It has been noted that students fail to understand that a fraction is not made up of two unrelated numbers but is actually a part-whole relationship. They fail to understand diagrams of fractions beyond what they see. Another factor is the way the students are taught. Current strategies do not emphasize on the need for students to ‘think outside the box’. For example, students must not rely only on the use of pie charts to understand fractions. Rather they should be able to infer, using their creativity, and so solve fraction problems. It has been noted that students are used to the circle diagrams to understand fractions, and so mathematics teachers must endeavor to increase the students’ exposure to other types of geometric models, for example rectangular shapes. Also, many text books ask students to shade circles to show their understanding of fractions. This then cultivate sin the students that all problems involving fractions can be solved using the circles. This is a dangerous and harmful mindset to have and it is one that contributes greatly to the difficulties that students experience when they meet fractions that are not common, for example, circles are not easily divided without the use of a protractor. Van de Walle. A mathematician suggested that another factor that adds to the students’ inability to understand fractions is the way fractions are read. A fraction figure is read from top to bottom, but the numerator (the top number) only makes sense if the denominator (the bottom number) is visualized. For instance, the ‘two’ in two thirds only makes sense when the third is shown as to be part of a whole. Therefore, students must be taught that the larger the number of the denominator, the smaller the fraction, and the smaller the denominator, the larger the fraction. Students should be educated on the reverse thinking that is normally applied to fractions. Another conceptual complexity that students have of fractions is when it comes to comparing whether fractions are equal. They normally have trouble because they see fractions as ‘representing double counts of unrelated numbers.’(Jigyel and Afanasaga-Fuata, 2007). The misunderstanding of fractions by students is also compounded by the mental images that they have of fractions. The normal teaching methods have trained the students to solve fractions that are common. However, when faced with unusual fractions, for example 3/89 x 93/230, the students are unsure as to how to proceed with solving this problem. Fractions, just like natural numbers, involve four operations-addition, subtraction, multiplication and division. Addition and subtraction and multiplication do not seem to pose as severe problems as the division of fractions. The normal approach to division of fractions is the “invert and multiply”. In this method the second fraction is inverted and then multiplied with the first fraction. There have been some criticisms for this tactic. Wu (1999) as cited in Fredua-Kwarteng and Ahia asserts that this approach misleads students into thinking that only the problems that they can visualize are the ones tat they can do. He also says that this approach does not exercise the students’ creativity because it only requires them to solve fraction problems that they can see or visualize rather than imagine; this then ill-prepares students when they are confronted with unusual fractions. There is new way of approaching the teaching of fractions as proposed by Kwarteng and Ahia, 2007.where they propose that division of fractions can b simplified by creating a common denominator where the denominators are different. For instance, 3/4 divided by 4/5.These fractions have different denominators. Therefore to convert the denominators, the numerators and denominators of the first fraction are multiplied by 5 and the second fraction by 4. This method has several advantages: Firstly, students can easily handle ordinary division of whole numbers. Secondly, the same skill of converting denominators is use in addition and subtraction of fractions, students get to practice that skill. Thirdly, students will not be required to learn by heart the common approach of “invert and multiply”. Fourthly, students gain a clear understanding of the interdependence between division as an operation and fractions as quotients. There is another approach to the division of fractions. Here, students apply algebraic algorithms to the fractions. It is similar to the ‘similar denominators’ approach but its main limitation is that it cannot be used by students without knowledge of algebraic manipulations. Hence, it is only suitable for students in grades 10 or 11. 3. Ways students can build knowledge Teachers should encourage their students to talk and write about their knowledge of fractions and their equivalents. This will strengthen their understanding of fractions along with providing valuable information to the teachers which will help them to gauge their students’ attitudes and performance.Students can build knowledge of fractions through fraction games. These games can include folding paper strips into several parts and students are asked to explain the divisions. These games test their understanding of how fractions are named. Fraction games also encourage students to think that fractions are fun and are not something to be feared. Also, fraction games expand their reasoning, subsequently leading the students to perform well in their study of Mathematics.(Harrison et al 1989). Another way that students can increase their knowledge of fractions is by using fraction walls. A fraction wall consists of en rows that are segmented into various fractions. Students are then required to complete the fraction wall using their paper strips to show the different fractions. This tests the students’ understanding of the equivalence of fractions and the sizes of fractions. It is important that the language used to teach students about fractions be consistent. The language of fractions describes how students will express fractions. The students should be asked to write in simple English the various fractions, for example, one seventh for 1/7. According to several studies, students normally approach fractions as what they can see. The ways that they have been taught do not usually encourage creativity when trying to understand fractions. The students are accustomed to the ‘pie chart’ method, and that is how their understanding of fractions is created. However, when faced with uncommon fractions, the students draw a blank. It has been suggested that teachers employ methods that will make the student think outside the box. This use of creativity is what will ensure the students actually learn to understand fractions that are both simple and complex. When the learning process is inhibited, it becomes difficult for students to internalize what they have been taught. The teacher must therefore ensure that students learn to creatively solve problems dealing with fractions. Another way that students can increase their knowledge of fractions is by doing tasks that help them ‘develop numerical relationships and strategies’ that are adaptable enough to be used in different circumstances (Pearn, 2007). Of course, a major problem that hinders students from learning fractions properly is their attitudes. Because they are accustomed to learning by rote the procedures, and applying these routines to practical situations is not always possible. Therefore they tend to develop negative attitudes towards fractions. This unfavorable attitude impedes their progress when performing mathematical examinations, and especially when it comes to fractions. Therefore, teachers must endeavor to change the students’ perception of fractions and their imagined ‘difficulty’. They can do this by encouraging discussion among students and also helping them to apply fractions to every day situations. This will give them the confidence that they will need to overcome their fear of fractions. It has been noted in various research studies that paying attention to the importance of the students understanding rational numbers and giving them the opportunity to use concrete materials to come up with imaginative situations improves their attitudes towards fractions and their operations. 4. Suggested approaches and strategies The most common method of teaching fractions used by teachers is where they perform several examples on the blackboard, and then explains the concepts used .Thereafter they give the students problems to work on. Normally, the students who find the topic problematic are then given assistance. In this method, students spend more time using textbooks as the source of their information and discussion is limited. The current model of teaching fractions to students is through geometric area models. These models are usually circles and sometimes rectangles are used. Rarely used are discrete models and fraction wall charts. The geometric approach in the teaching of fractions Concrete process oriented methods These approaches typically use various simple concrete materials to set problems, that when the students use the concrete materials to solve, will aid in the understanding of the mathematical concepts entailed. In these methods, teachers first introduce a discussion of a problem, and the class is divided into groups. These groups then attempt to solve the problem using concrete materials. The teacher helps the students not by giving direct answers but rather by giving hints that the students use to solve out the problem on concrete materials. The advantage of this method is that the class discussions provide a means of discovering the level of the students’ comprehension of fractions. Also, the method allows students to work at their own levels of mental ability. This method pays attention to the importance of students’ understanding of the logic behind rational numbers. Furthermore, the discussions enable students to think about and digest what they have learnt. The use of concrete materials helps the students to organize their concepts and understand better. Lastly, the investigational aspect of this approach helps the students to relate new ideas to already existing representations of fractions and their uses. The measurement model is a model that is created to help teachers to point out mathematical understanding of their pupils and to come up with activities that will aid all students to improve at their level of understanding. In this model, students fold strips of paper into fractional parts and then use those strips to note fractions divisions and number lines. Students can compare and arrange fractions in order. The advantage of this model is that it uses folded paper strips to complete the fraction wall and then mark the suitable positions for fractions on number lines. These actions help students to cultivate a grasp of fractions, rather than depend on rules and procedures without a clear comprehension. This model emphasizes the need to develop in students the language of fractions and also the skill to move between common language and fractional symbols. (Pearn, 2007).The students should be able to decipher written words and translate that into the correct mathematical symbols. This paper recommends that teachers of fractions use the process oriented methods There are three ways in which the students’ understanding of fractions can be measured. First, the students must be able to add and subtract fractions using concrete materials, for example pictures, oranges and the like. Secondly, they should be able to recognize and write equal fractions when presented with pictographic aid. Thirdly, students should be able to translate fractions into decimals and vice versa. In group discussions the students should focus on the problems that they have been given. Thirdly, the teacher should then assist the students to infer information and us their creativity to solve the problems. It has been noted that for students to fully comprehend the concepts of fractions, they should be able to imagine the fractions as well as visualize them. Process oriented methods include the measurement model. Here students are given concrete materials to use, i.e. paper. The use of concrete materials helps students to internalize the lessons. Process oriented teaching strategies have been found to considerably enhance accomplishment in students’ study of fractions and also to expand their ways of thinking towards a more positive mind set as regards fractions. 5. Conclusion Fractions are important in our day to day lives, and students must be made aware of this. It is important that they see the various areas that fractions are used in, for example, in the medical profession, in engineering, in architecture and even in the home. By using practical examples in the teaching of fractions, students will be able to relate the fractions that they learn in class and hence completely internalize them. This poses benefits for the students because their success in their mathematical studies will be guaranteed. It is of great importance to nurture the students’ understanding of the concepts of fractions, which include the part-whole relationship, the number of sections and the size of the sections, using various representations. . For students, fractions are understood in two ways: the concept definition and the concept image. Students are found to use the definition of a concept to recognize the characteristics of the concept rather than the concept itself (Tall, 1991 as cited in Kwarteng and Ahia 2007).Teachers should endeavor to understand the basis at which students approach fractions and should then try to positively influence them. It must be remembered that even teachers have their conceptual difficulties when teaching and they should be aware of them. REFERENCES Fredua-Kwarteng, E. and Ahia, F (2007). Understanding Division of Fractions: An Alternative view Harrison, B et al (1989) Allowing for Student Cognitive Levels in the Teaching of Fractions and Ratios Jigyel, K and Afamasaga-Fuata’I, K (2007). Students’ conceptions of models of fractions Pearn, C.A (2007). Using Paper folding, fraction walls and number lines to develop Understanding of fractions for students from Years 5-8 Yiu-Kwong, M (2008). A study of two-term unit fraction expansions via geometric approach. Read More