Reshma Gangadeen (Group5 ) Why fractions are difficult to grasp? Problems with fractions often develop from the fact that they are different from natural numbers in that they are relative rather than a fixed amount - the same fraction might refer to different quantities and different fractions may be equivalent (Pennant& Woodham, n.d). This is the first aspect that is very difficult for a young child to understand. Therefore in order for students to develop a better understanding of fractions they need to be totally clear on what a fraction means. A fraction is a part of a “whole”, whole what? Fractions can refer to objects, quantities or shapes, which can make the term fraction far more difficult for students to understand.
In order to develop students understanding and then simplify fractions, children need to discover many representations and uses over a substantial period of time. Children should have experiences of objects, shapes and amounts in equal amount and they should also experience of the whole being something other than one. (Pennant & Woodham,n.d) this will more than likely make the concept more clear to them.
Based on the above research I think the reason why we find fractions difficult to learn as well as teach is because we are not taking the correct approach to teaching it. We need to use a number of strategies as well as expose students to all different types and forms of fractions. We need to show/teach the topic to in ways that will be meaningful to them and with examples that they can identify with (using real life situations and examples). Therefore once the approach is changed and students can better relate to what they are being exposed to, the learning process should become easier and students will be able to grasp the concept/s they are encountering.
According to Pitkethly and Hunting (1996), fractions continue to present problems and difficulties for children in primary schools.
Teaching fractions is indeed a challenge at primary school because children tend to make all sorts of errors not only in the computation of fractions, but also in the basic concept.
Orton (1992), states that the concept of fractions should be developed over a long period of time, during which time children experience the different meanings of fractions in a variety of situations.
It must be noted that children encounter fractions and fraction-related concepts in both real-life and in classroom situations. Therefore, teachers should build the concept of fractions on the children's personal experiences.
Generally speaking, the experience with fractions at the primary school level for a lot of children can be simply described as an inductive process. When fractions are to be taught it is done as a procedure. For example, students may learn to order fractions by looking at the denominator and saying that the larger the denominator the smaller the fraction. Moreover, this procedure is followed throughout ordering fractions regardless of the value of the numerator.
All in all, the teaching and learning of fractions at the lower level is an experience that can lead to misconceptions.
Reference: www. merga.net.au/...RR_yusof.pdf
Done by: C. Bisram, R. Simbhoo, K. Khan, C. Ramdhanee, E. Sucre.
Fazida Yathali, Reshma Gangadeen, Ornella Oudai, Abdur Mohammed (Group 5) As many teachers and parents know, learning the various fraction operations can be difficult for many children. It's not the concept of a fraction that is difficult - it is the various operations: addition, subtraction, multiplication, division, comparing, simplifying, etc. of fractions (“Teaching Fractions,” 2015). The simple reason that the different operations are difficult to grasp is because of how it is taught and how many rules that there are to be remembered. For example a rule in conversion of fractions to decimals is to remember to divide using long division. Another rule in fractions is when converting a mixed number to a fraction, multiply the whole number part by the denominator and add the numerator to get the numerator then, use the common denominator as in the fractional part of the mixed number. Those are just two examples and the list goes on. Students try to memorize these rules, which they may soon forget, and don’t try to understand the concepts. Then the whole idea of fraction becomes a meaningless jungle. Suffice it to say that teachers need to teach in such a way to get the students to reach a point of relational understanding (Skemp, 1986). However, it is not only difficult for students to learn but also for teachers to deliver this topic of fractions. Therefore, to make teaching fractions a simpler task, instead of teachers presenting a rule, they should try to use visual models or manipulatives and this way it becomes meaningful concrete experience for the students and not just a number on top of number without meaning. I strongly believe that only then will students be able to understand. They may eventually be able to estimate an answer before working it out, discuss the reason behind the answer and perform many of the problems mentally without the application of any “rule”.
References:
Teaching Fractions Why are Fractions so Difficult to Learn. (2015, February 26). Retrieved from http://www.homeschoolmath.net/teaching/teaching-fractions.php
Skemp, R.R (1986). The Psychology of Learning Mathematics (2nd ed.). London: Penguin Books.
Group Members: Merisa Jhagmohan, Tamara Charles Apping, Lisa Ramdath, Andre Cadette, Cassandra Mohammed – Ali
Fractional concepts are important building blocks for the students of primary school level and secondary school level. Conceptually based instruction of fractions requires teachers to have a complete understanding of the subject matter. Several researchers (e.g., Ball, 1990; Shulman, 1986; Wilson, Shulman, & Richert, 1987) have proposed theories about teacher knowledge characteristics and structure. In the area of mathematics, Hill, Schilling, and Ball (2004) have extended Shulman's original ideas about pedagogical content knowledge and have developed a model for mathematics' teacher knowledge referred to as mathematics knowledge for teaching (MKT). In their model, the three knowledge domains most central to mathematics teaching are common knowledge of mathematics, specialized knowledge of content, and knowledge of students and their ways of thinking about the content. Hill and associate's model provides a theoretical base for what primary school teachers need to know to teach fractions. Common knowledge is the knowledge that a mathematically educated adult, not necessarily a teacher, needs to possess to provide correct mathematical solutions. Specialized knowledge of content is the possession of mathematical knowledge and skills such as being able to explain why an algorithm works or being able to provide students with multiple representations addressing diverse learning styles. Although most of the fractional operations; addition, subtraction, and multiplication are covered in primary school, they are often revisited in secondary school. Research suggests that students have a procedural knowledge of fractional operations rather than an understanding of underlying concepts (Mack, 1990).
Reference: Ball, D. L. (1993). Halves, pieces, and twoths: Constructing and using representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, &
T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 157–195). Hillsdale, NJ: Erlbaum. Mack, N. K. (1990). Learning fractions with understanding. Journal for Research in Mathematics Education, 21, 16–32.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4–14.
Wilson, S. M., Shulman, L. S., & Richert, A. (1987). 150 different ways of knowing: Representations of knowledge in teaching. In J. Calderhead (Ed.), Exploring teachers' thinking (pp. 104–124). London: Cassell Education.
Group Members: Merisa Jhagmohan, Tamara Charles - Apping, Lisa Ramdath, Andre Cadette, Cassandra Mohammed – Ali.
NCTM (n. d.) asserted that "multiplication by fractions and decimals can be challenging for primary level students if experiences with multiplication by whole numbers have led them to believe that multiplication makes “bigger”. To communicate the effects of multiplication by numbers less than 1, teachers can use concrete models such as manipulatives. Furthermore, even if students are capable of solving problems such as John has 4 cakes and Ann has 2/3 as many cakes as John, students think of these problem situations as multiplication rather than division because the result is smaller (Taber, 2002). Therefore, Taber suggested that instruction of multiplication with fractions should relate to multiplication with whole numbers while reconceptualizing students' understanding of whole-number multiplication to include fractions as multipliers (Taber). Teachers need to have the necessary specialized knowledge, such as knowledge of a variety of representations including concrete models and real-world problems, to help students visualize or relate as they transition from multiplication by whole numbers to multiplication by fractions (Taber).
References:
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Taber, S. B. (2002). Go ask Alice about multiplication of fractions. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook (pp. 61–71). Reston, VA: National Council of Teachers of Mathematics.
Group members: RishmattieMaharaj, Christine Jainth, AngallieMaraj, Maya Dass
Fractions: difficult but crucial in mathematics learning Quantities represented by natural numbers are easily understood. We can count and say how many oranges are in a bag. But fractions cause difficulty to most people because they involve relations between quantities. For example, what is 1/2? One half of what? The relative nature of fractions is a source of difficulty for pupils. It requires that they realise that the same fraction may refer to different quantities (1/2 of 8 and 1/2 of 12 are different) and that different fractions may be equivalent because they refer to the same quantity (for example1/3 and 3/9,). It is not possible for pupils to make further progress in mathematics or to take advanced courses in secondary school without a sound grasp of the relative nature of rational numbers. Research on fractions has shown that many of the mistakes which pupils make when working with fractions can be seen as a consequence of their failure to understand that natural and rational numbers involve different ideas. One error that pupils make with fractions is to think that, for example, 1/3 of a cake is smaller than 1/5 because 3 is less than 5. Yet most children readily recognise that a cake shared among three children gives bigger portions than the same cake shared among five children. Because children do show good insight into some aspects of fractions when they are thinking about division, mathematics educators have begun to investigate whether these situations could be used as a starting point for teaching fractions. Fractions are difficult at primary level but it is also crucial in mathematics learning therefore we as teachers should be mindful of this. References Teaching and Learning Research Programme. (2006, January 1). Retrieved February 28, 2015, from http://www.tlrp.org/pub/documents/no13_nunes.pdf https://books.google.tt/books?id=5aaTAgAAQBAJ&pg=PR66&dq=difficult+of+fractions&hl=en&sa=X&ei=LSPxVOH5JZbbsATvioDgAw&ved=0CCoQ6AEwAw#v=onepage&q=difficult%20of%20fractions&f=false
Group members: Rishmattie Maharaj, Christine Jainth, Angallie Maraj, Maya Dass
Quantities represented by natural numbers are easily understood. We can count and say how many oranges are in a bag. Fractions cause difficulty to most people because they involve relations between quantities for example, what is 1/2? One half of what? The relative nature of fractions is a source of difficulty for pupils. Students often realise that the same fraction may refer to different quantities (1/2 of 8 and 1/2 of 12 are different) and that different fractions may be equivalent because they refer to the same quantity (for example1/3 and 3/9,). It is not possible for pupils to make further progress in mathematics or to take advanced courses in secondary school without a sound grasp of the relative nature of rational numbers. Research on fractions has shown that many of the mistakes that pupils make when working with fractions can be seen as a consequence of their failure to understand that natural and rational numbers involve different ideas. One error that pupils make with fractions is to think that, for example, 1/3 of a cake is smaller than 1/5 because 3 is less than 5. Yet most children readily recognise that a cake shared among three children gives bigger portions than the same cake shared among five children. However, children do show good insight into some aspects of fractions when they are thinking about division. Mathematics educators have begun to investigate whether these situations could be used as a starting point for teaching fractions. Fractions are difficult at primary level but it is also crucial in mathematics learning therefore we as teachers should be mindful of this.
References Teaching and Learning Research Programme. (2006, January 1). Retrieved February 28, 2015, from http://www.tlrp.org/pub/documents/no13_nunes.pdf
Why fractions are difficult to teach and learn in the Early Levels
Our group found it useful to use the following model from Derek Haylock and Anne Cockburn (Haylock and Cockburn; 1989) to consider the different mathematical elements that need to be experienced and connected in order to create full understanding of concepts. Haylock and Cockburn suggest that effective learning takes place when the learner makes cognitive connections. Considering a particular example in early fractions. Two children are cooking, filling a tray of 12 cake cases. They are told they can fill half each. One child looks at the tray and says, “We can do two lines each”. The other child looks at the lines and says “That’s six because three and three is six, like on a dice”. The children fill six cake cases each. The cooking is the context, the tray and dice the images, the language of fractions, division and multiplication is used and there is the opportunity to model both 12 x 1/2 = 6, 12 ÷2 = 6 and 6x 2 = 12. Problems can arise when not all the four elements are experienced or, if they are all experienced, but they are not connected in a meaningful way. The role of classroom dialogue is to help the children make the connections themselves. This dialogue can take the form of teacher questioning, children questioning, talk between children and explanation of points of view. The verbal accompaniment to the children’s experiences is what allows them to frame their understanding. One of the things we need to ensure, as teachers, is that children are given a variety of experiences that allow them to engage with fractions as both the names of numbers and also as operators.
Reference
Haylock, D., & Cockburn, A. (1989, January 1). Why do fractions and decimals seem difficult to teach and learn? Retrieved March 4, 2015, from http://www.annery-kiln.eu/gaps-misconceptions/fractions/why-fractions-difficult.html
Group members: Rishmattie Maharaj, Christine Jainth, Angallie Maraj, Maya Dass
Difficulties with Fractions
The understanding of the concepts of fractions is very important in understanding equivalent fractions. Students have considerable difficulties with fraction equivalence, the notion that different fractions can represent the same amount (Bana, Farrell, & McIntosh, 1997; Pearn, Stephens, & Lewis, 2003 as cited in Anderson & Wong, 2007, p. 1). Throughout our life, people have known that one number represents one quantity. Kamii & Clark (1995) reported that it is vital to understand equivalent fractions because it develops a base for understanding addition and subtraction of fractions and allows students to compare and order fractions. Failure in understanding the concept of equivalent fraction in primary school can cause students to face difficulties in higher level education especially in the understanding of algebra.
Reference
Anderson, J., & Wong, M. (2007). Teaching Common Fractions In Primary School: Teachers‟ Reaction to a New Curriculum. Proceedings of the Australian Association for Research in Education’s 36th Annual International Education Research Conference, Volume 1. (pp. 1-13). Retrieved January 29, 2009, from http://www.aare.edu.au/06pap/and06181.pdf
Group members Fazida Yathali, Reshma Gangadeen, Ornella Oudai, Abdur Mohammed
The importance of fractions
Students in our primary schools do not necessarily see the importance of factions at their young, tender age. This now becomes a requirement of the teachers to show their students the importance of fraction. Fractions are the pillars for almost every other math topic there is and if the students do not see the connections then it would be very problematic for the child's future in mathematics. Fractions enable students to grasp the nature of numbers and their concepts. When fractions are misinterpreted it will interfere with the ability of the child to quickly understand mathematical processes later on in life such as algebra. This is only looking at an angle from school however we meet fractions everyday in our lives from buying groceries to even cooking food fractions play a very important role for everyone.
Reference
Hanich, L. (2009, July 7). Why are fractions so important? Retrieved March 7, 2015, from http://www.svsd.net/cms/lib5/PA01001234/Centricity/Domain/1/theparentpage/articles2/161.pdf
Teaching and learning maths/fractions at an early level can often be difficult for students to understand primarily because of the the way it is introduced to children. As many teachers and parents know, learning the various fraction operations can be difficult for many children. It's not the concept of a fraction that is difficult - it is the various operations: addition, subtraction, multiplication, division, comparing, simplifying, etc. of fractions The simple reason why learning operations can prove difficult for many students is the way they are typically taught. Just look at the amount of rules there are to learn about fractions! 1. Fraction addition - common denominators Add the numerators, and use the common denominator 2. Fraction addition - different denominators First find a common denominator by taking the least common multiple of the denominators. Then convert all the addends to have this common denominator. Then add using the rule number 1. 3. Finding equivalent fractions Multiply both the numerator and denominator by a same number. 4. Convert a mixed number to a fraction Multiply the whole number part by the denominator and add the numerator to get the numerator. Use the common denominator as in the fractional part of the mixed number. 5. Convert an improper fraction to a mixed number Divide the numerator by the denominator to get the whole number part. The remainder will be the numerator of the fractional part. Denominator is the same. 6. Simplifying fractions Find the (greatest) common divisor of the numerator and denominator, and divide both by it. 7. Fraction multiplication Multiply the numerators and the denominators. 8. Fraction division Find the reciprocal of the divisor, and multiply by it. 9. Comparing fractions Convert the fractions so they have a common denominator. Then compare the numerators. 10. Convert fractions to decimals Divide using long division or a calculator.
If students simply try to memorize these rules without knowing where they came from, the rules will probably seem like a meaningless jungle. They probably won't seem to connect with anything about the operation, but instead work like "magic": you multiply, divide, and do various things with the numerators and denominators to come up with the answer. Students can then become blind followers of the rules, tossing numbers here and there, calculating this and that - and getting answers without having any idea if they are reasonable or not. Besides, it is quite easy to forget these rules or misremember them - especially after 5-10 years.
The solution: manipulatives and visual models Instead of merely presenting a rule, a better way is to use visual models or manipulatives during the study of fraction arithmetic. That way fractions become something concrete to the student, and not just a number on top of another without a meaning. The student will be able to estimate the answer before calculating, evaluate the reasonableness of the final answer, and perform many of the simplest operations mentally without knowingly applying any "rule." Now, typical textbooks DO show visual models for fractions, and they DO show one or two examples of how a certain rule connects with a picture. But that is not enough! We need to have children solve lots of problems using either visual models or fraction manipulatives. Another way is to ask them to DRAW fraction pictures for the problems. That way the students will form a mental visual model and can think through the pictures.
Fractions is undoubtedly one of the more challenging topics to teach to young learners. Teachers are often forced to try "different approaches" to teaching fractions because students seem unable to grasp the concepts. Actually, students have a knowledge of the concepts of Fractions, however, problems begin when they are required to understand the operations for working with Fractions.
There is a growing debate regarding why fractions pose so may challenges to young learners. One argument is that students struggle with Fractions because of HOW it was taught to them. This perspective places the blame squarely on the educators; in other words, the reason students struggle with Fractions is because of "us".
Leah rightly referenced a solution to this dilemma, "Instead of merely presenting a rule, a better way is to use visual models or manipulatives during the study of fraction arithmetic." Students need to be able to make sense of what they are learning. If students are mainly presented with a host of rules and formulae, they would not develop a working knowledge or understanding of Fractions. As with any other area of teaching and learning, students must be taken from the concrete to the abstract in order to help develop their understanding of a subject or topic.
In light of this, teachers need to pay closer attention to the previous knowledge of their learners as this will be the foundation on which they build. A child's first introduction to fractions is also of critical importance. It must be done correctly the first time in order to maximize the learning experience for the child. If fractions were not introduced properly, the child would not be successful at it. The child may also develop a mental block towards Fractions because of the initial classroom experience.
Successful strategies for teaching Fractions to young children have been developed. Teachers now have the privilege of being taught the Science of teaching young children Mathematics. While different methods may be presented, two approaches stand out:-
(1) If students are to overcome challenges with Fractions, they must be allowed to discover the meaning of the relationships. (2) Students must be allowed to work with relevant concrete or tactile stimuli in order to make sense of Fractions.
Hence, the word "challenges" with reference to Fractions and children's learning should become increasingly obsolete as teachers now have the tools and knowledge of how to effectively deliver their instructions.
Group members: Rishmattie Maharaj, Christine Jainth, Angallie Maraj, Maya Dass
The Challenge
Fractions are the earliest topic in school mathematics where educators agree that students fail and teachers struggle to teach. “Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra.” (National Math Advisory Panel, 2008.) This challenge is understandable as fractions present major conceptual leaps for students. Consider these factors:
Fractions can describe many different things, including parts of a whole, parts of a set, time, and length.
Sophisticated reasoning is required to evaluate any fraction. Students must analyze the relationship between two numbers in order to understand a single value, and recognize that the real value of a fraction is dependent upon the unit, or whole, of which it is a part.
Fractions operations can be multi-step and abstract. Addition and subtraction can require multiple steps, while multiplication and division are too abstract for many people.
Fractions present a plethora of new terms for students to master. Terminology like numerator, denominator, equivalent, common, uncommon, proper, and improper can confound students, especially students who are struggling with reading.
Our group came up with two solutions to help teachers and students with this problem. A solution to this problem is by using visual models to progress from concrete to abstract. According to the NCTM Standards, educators have consistently agreed on the value of moving from concrete manipulative, to pictorial representations, to abstract numbers. Teachers should start their lessons with the visual models and the language, and then lead students to understanding fractions and solving number problems. Another solution to this challenge was to provide concentrated time with supplemental lessons. Instructional time is a key to success, and every instructional minute is precious. Students can successfully complete as many problems in 30 minutes. Students experience success with instruction that unfolds logically and adapts to their level. The concentration of successful interactions builds synaptic connections and sets the stage for success.
Group #5 Members: Fazida Yathali, Ornella Oudai, Reshma Gangadeen, Abdur Mohammed. Learning and Teaching Fractions A fraction is part of a whole. It's less than 1 whole thing, but more than 0. We use fractions all the time in real life. Have you ever ordered a quarter-pound burger? Or noticed that your gas tank is half full? Both of these are fractions of the whole amount, a whole pound of meat, or a whole tank of gas. Every fraction has two parts: a top number and a bottom number. In math terms, these are called the numerator and the denominator. As long as you remember what each number means, you can understand any fraction. The top number, or numerator, refers to a certain number of those parts. It lets us know how much we're talking about. There are many ways how to teach fractions, and the success or failure of the student to understand fractions will depend on the teaching method used. It's been said that if a student understands fractions, then they can understand any mathematics concept. It is then very important for every math teacher to know how to teach fractions in the most approachable way possible. When deciding on a method of how to teaching fractions, we need to use fractional analogies that the student will immediately recognize. Thus enters the pizza as the perfect instrument needed to teach the concept of the fraction. All students learn differently therefore several methods of teaching should be used when teaching fractions. How to Teach Fractions. (1998). Retrieved March 13, 2015, from http://www.mathgoodies.com/articles/teach_fractions.html Free Fractions Tutorial at GCFLearnFree. (1998). Retrieved March 13, 2015, from http://www.gcflearnfree.org/fractions
Group members: Rishmattie Maharaj, Christine Jainth, Angallie Maraj, Maya Dass
The Essential Components of Understanding Fractions
The topic of fractions is an integral part of the elementary school mathematics curriculum, yet it is one of the most difficult for students to master. Understanding and being able to use fractions is essential for mathematics success; however, traditional teaching methods often focus on procedures and set of rules rather than deep conceptual understanding that is necessary for fraction success. “Children are bound to find fractions computations arbitrary, confusing and easy to mix up unless they receive help understanding what fractions and fraction operations mean” (Siebert & Gaskin, 2006, p. 394). Teachers feel that teaching fractions is a challenge because they must consider what will help to deepen students’ understanding (Yoshida & Shinmachi, 1999).
In order to assist children in developing a deep conceptual understanding of fractions, teachers must realize that fractions are not simply “algorithms to be taught (Faulkner, 2009, p. 28). Teachers should understand that fractions are in fact a vital part of each component of number sense.
By considering how fractions are directly related to each aspect of number sense, teachers will be better able to understand the important mathematical bases for fractions and be able to use these as a foundation for teaching fractions to students in a way that supports linking fractions to other aspects of mathematics, thus deepening conceptual understanding. In addition to understanding how fractions are integrated into number sense, teachers should use a number of methods for increasing conceptual understanding, such as representations, models, manipulatives, and games; yet make sure that they are developmentally appropriate, relevant to real-life, and do not cause misconceptions.
One area of improvement could be on how teachers engage students. Students should be shown how fractions apply to their personal lives. This could make the lesson more motivating and successful for the students. Many teachers do not make the connections from the content to real life applications. Many of the fraction lessons taught are not engaging or actively involve the students. Instead, they consisted of the teacher providing examples and the students practicing.
References
Brizuela, B.M. (2006). Young children’s notations for fractions. Educational Studies in Mathematics, 62(3), 281-305.
Chick, C., Tierney, C., & Storeygard, J. (2007). Seeing Students' Knowledge of Fractions:
This comment has been removed by the author.
ReplyDeleteReshma Gangadeen (Group5 )
ReplyDeleteWhy fractions are difficult to grasp?
Problems with fractions often develop from the fact that they are different from natural numbers in that they are relative rather than a fixed amount - the same fraction might refer to different quantities and different fractions may be equivalent (Pennant& Woodham, n.d). This is the first aspect that is very difficult for a young child to understand. Therefore in order for students to develop a better understanding of fractions they need to be totally clear on what a fraction means. A fraction is a part of a “whole”, whole what? Fractions can refer to objects, quantities or shapes, which can make the term fraction far more difficult for students to understand.
In order to develop students understanding and then simplify fractions, children need to discover many representations and uses over a substantial period of time. Children should have experiences of objects, shapes and amounts in equal amount and they should also experience of the whole being something other than one. (Pennant & Woodham,n.d) this will more than likely make the concept more clear to them.
Based on the above research I think the reason why we find fractions difficult to learn as well as teach is because we are not taking the correct approach to teaching it. We need to use a number of strategies as well as expose students to all different types and forms of fractions. We need to show/teach the topic to in ways that will be meaningful to them and with examples that they can identify with (using real life situations and examples). Therefore once the approach is changed and students can better relate to what they are being exposed to, the learning process should become easier and students will be able to grasp the concept/s they are encountering.
References
Pennant, J, and Woodham, L.,(n.d). Understanding Fractions. Retrieved from:
http://nrich.maths.org/10496 date: 2/19/15
According to Pitkethly and Hunting (1996), fractions continue to present problems and difficulties for children in primary schools.
ReplyDeleteTeaching fractions is indeed a challenge at primary school because children tend to make all sorts of errors not only in the computation of fractions, but also in the basic concept.
Orton (1992), states that the concept of fractions should be developed over a long period of time, during which time children experience the different meanings of fractions in a variety of situations.
It must be noted that children encounter fractions and fraction-related concepts in both real-life and in classroom situations. Therefore, teachers should build the concept of fractions on the children's personal experiences.
Generally speaking, the experience with fractions at the primary school level for a lot of children can be simply described as an inductive process. When fractions are to be taught it is done as a procedure. For example, students may learn to order fractions by looking at the denominator and saying that the larger the denominator the smaller the fraction. Moreover, this procedure is followed throughout ordering fractions regardless of the value of the numerator.
All in all, the teaching and learning of fractions at the lower level is an experience that can lead to misconceptions.
Reference: www. merga.net.au/...RR_yusof.pdf
Done by: C. Bisram, R. Simbhoo, K. Khan, C. Ramdhanee, E. Sucre.
ReplyDeleteFazida Yathali, Reshma Gangadeen, Ornella Oudai, Abdur Mohammed (Group 5)
As many teachers and parents know, learning the various fraction operations can be difficult for many children. It's not the concept of a fraction that is difficult - it is the various operations: addition, subtraction, multiplication, division, comparing, simplifying, etc. of fractions (“Teaching Fractions,” 2015). The simple reason that the different operations are difficult to grasp is because of how it is taught and how many rules that there are to be remembered. For example a rule in conversion of fractions to decimals is to remember to divide using long division. Another rule in fractions is when converting a mixed number to a fraction, multiply the whole number part by the denominator and add the numerator to get the numerator then, use the common denominator as in the fractional part of the mixed number. Those are just two examples and the list goes on. Students try to memorize these rules, which they may soon forget, and don’t try to understand the concepts. Then the whole idea of fraction becomes a meaningless jungle. Suffice it to say that teachers need to teach in such a way to get the students to reach a point of relational understanding (Skemp, 1986).
However, it is not only difficult for students to learn but also for teachers to deliver this topic of fractions. Therefore, to make teaching fractions a simpler task, instead of teachers presenting a rule, they should try to use visual models or manipulatives and this way it becomes meaningful concrete experience for the students and not just a number on top of number without meaning. I strongly believe that only then will students be able to understand. They may eventually be able to estimate an answer before working it out, discuss the reason behind the answer and perform many of the problems mentally without the application of any “rule”.
References:
Teaching Fractions Why are Fractions so Difficult to Learn. (2015, February 26). Retrieved from
http://www.homeschoolmath.net/teaching/teaching-fractions.php
Skemp, R.R (1986). The Psychology of Learning Mathematics (2nd ed.). London: Penguin Books.
Group Members: Merisa Jhagmohan, Tamara Charles Apping, Lisa Ramdath, Andre Cadette, Cassandra Mohammed – Ali
ReplyDeleteFractional concepts are important building blocks for the students of primary school level and secondary school level. Conceptually based instruction of fractions requires teachers to have a complete understanding of the subject matter. Several researchers (e.g., Ball, 1990; Shulman, 1986; Wilson, Shulman, & Richert, 1987) have proposed theories about teacher knowledge characteristics and structure. In the area of mathematics, Hill, Schilling, and Ball (2004) have extended Shulman's original ideas about pedagogical content knowledge and have developed a model for mathematics' teacher knowledge referred to as mathematics knowledge for teaching (MKT). In their model, the three knowledge domains most central to mathematics teaching are common knowledge of mathematics, specialized knowledge of content, and knowledge of students and their ways of thinking about the content. Hill and associate's model provides a theoretical base for what primary school teachers need to know to teach fractions. Common knowledge is the knowledge that a mathematically educated adult, not necessarily a teacher, needs to possess to provide correct mathematical solutions. Specialized knowledge of content is the possession of mathematical knowledge and skills such as being able to explain why an algorithm works or being able to provide students with multiple representations addressing diverse learning styles. Although most of the fractional operations; addition, subtraction, and multiplication are covered in primary school, they are often revisited in secondary school. Research suggests that students have a procedural knowledge of fractional operations rather than an understanding of underlying concepts (Mack, 1990).
Reference:
Ball, D. L. (1993). Halves, pieces, and twoths: Constructing and using representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, &
T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 157–195). Hillsdale, NJ: Erlbaum.
Mack, N. K. (1990). Learning fractions with understanding. Journal for Research in Mathematics Education, 21, 16–32.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15, 4–14.
Wilson, S. M., Shulman, L. S., & Richert, A. (1987). 150 different ways of knowing: Representations of knowledge in teaching. In J. Calderhead (Ed.), Exploring teachers' thinking (pp. 104–124). London: Cassell Education.
Group Members: Merisa Jhagmohan, Tamara Charles - Apping, Lisa Ramdath, Andre Cadette, Cassandra Mohammed – Ali.
ReplyDeleteNCTM (n. d.) asserted that "multiplication by fractions and decimals can be challenging for primary level students if experiences with multiplication by whole numbers have led them to believe that multiplication makes “bigger”. To communicate the effects of multiplication by numbers less than 1, teachers can use concrete models such as manipulatives. Furthermore, even if students are capable of solving problems such as John has 4 cakes and Ann has 2/3 as many cakes as John, students think of these problem situations as multiplication rather than division because the result is smaller (Taber, 2002). Therefore, Taber suggested that instruction of multiplication with fractions should relate to multiplication with whole numbers while reconceptualizing students' understanding of whole-number multiplication to include fractions as multipliers (Taber). Teachers need to have the necessary specialized knowledge, such as knowledge of a variety of representations including concrete models and real-world problems, to help students visualize or relate as they transition from multiplication by whole numbers to multiplication by fractions (Taber).
References:
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Taber, S. B. (2002). Go ask Alice about multiplication of fractions. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook (pp. 61–71). Reston, VA: National Council of Teachers of Mathematics.
Group members: RishmattieMaharaj, Christine Jainth, AngallieMaraj, Maya Dass
ReplyDeleteFractions: difficult but crucial in mathematics learning
Quantities represented by natural numbers are easily understood. We can count and say how many oranges are in a bag. But fractions cause difficulty to most people because they involve relations between quantities. For example, what is 1/2? One half of what? The relative nature of fractions is a source of difficulty for pupils. It requires that they realise that the same fraction may refer to different quantities (1/2 of 8 and 1/2 of 12 are different) and that different fractions may be equivalent because they refer to the same quantity (for example1/3 and 3/9,). It is not possible for pupils to make further progress in mathematics or to take advanced courses in secondary school without a sound grasp of the relative nature of rational numbers.
Research on fractions has shown that many of the mistakes which pupils make when working with fractions can be seen as a consequence of their failure to understand that natural and rational numbers involve different ideas. One error that pupils make with fractions is to think that, for example, 1/3 of a cake is smaller than 1/5 because 3 is less than 5. Yet most children readily recognise that a cake shared among three children gives bigger portions than the same cake shared among five children. Because children do show good insight into some aspects of fractions when they are thinking about division, mathematics educators have begun to investigate whether these situations could be used as a starting point for teaching fractions.
Fractions are difficult at primary level but it is also crucial in mathematics learning therefore we as teachers should be mindful of this.
References
Teaching and Learning Research Programme. (2006, January 1). Retrieved February 28, 2015, from http://www.tlrp.org/pub/documents/no13_nunes.pdf
https://books.google.tt/books?id=5aaTAgAAQBAJ&pg=PR66&dq=difficult+of+fractions&hl=en&sa=X&ei=LSPxVOH5JZbbsATvioDgAw&ved=0CCoQ6AEwAw#v=onepage&q=difficult%20of%20fractions&f=false
Group members: Rishmattie Maharaj, Christine Jainth, Angallie Maraj, Maya Dass
ReplyDeleteQuantities represented by natural numbers are easily understood. We can count and say how many oranges are in a bag. Fractions cause difficulty to most people because they involve relations between quantities for example, what is 1/2? One half of what? The relative nature of fractions is a source of difficulty for pupils. Students often realise that the same fraction may refer to different quantities (1/2 of 8 and 1/2 of 12 are different) and that different fractions may be equivalent because they refer to the same quantity (for example1/3 and 3/9,). It is not possible for pupils to make further progress in mathematics or to take advanced courses in secondary school without a sound grasp of the relative nature of rational numbers.
Research on fractions has shown that many of the mistakes that pupils make when working with fractions can be seen as a consequence of their failure to understand that natural and rational numbers involve different ideas. One error that pupils make with fractions is to think that, for example, 1/3 of a cake is smaller than 1/5 because 3 is less than 5. Yet most children readily recognise that a cake shared among three children gives bigger portions than the same cake shared among five children. However, children do show good insight into some aspects of fractions when they are thinking about division. Mathematics educators have begun to investigate whether these situations could be used as a starting point for teaching fractions.
Fractions are difficult at primary level but it is also crucial in mathematics learning therefore we as teachers should be mindful of this.
References
Teaching and Learning Research Programme. (2006, January 1). Retrieved February 28, 2015, from http://www.tlrp.org/pub/documents/no13_nunes.pdf
This comment has been removed by the author.
ReplyDeleteGroup 5
ReplyDeleteMembers: Fazida Yathali, Abdur Mohammed, Reshma Gangadeen, Ornella Oudai.
Why fractions are difficult to teach and learn in the Early Levels
Our group found it useful to use the following model from Derek Haylock and Anne Cockburn (Haylock and Cockburn; 1989) to consider the different mathematical elements that need to be experienced and connected in order to create full understanding of concepts. Haylock and Cockburn suggest that effective learning takes place when the learner makes cognitive connections. Considering a particular example in early fractions. Two children are cooking, filling a tray of 12 cake cases. They are told they can fill half each. One child looks at the tray and says, “We can do two lines each”. The other child looks at the lines and says “That’s six because three and three is six, like on a dice”. The children fill six cake cases each. The cooking is the context, the tray and dice the images, the language of fractions, division and multiplication is used and there is the opportunity to model both 12 x 1/2 = 6, 12 ÷2 = 6 and 6x 2 = 12.
Problems can arise when not all the four elements are experienced or, if they are all experienced, but they are not connected in a meaningful way. The role of classroom dialogue is to help the children make the connections themselves. This dialogue can take the form of teacher questioning, children questioning, talk between children and explanation of points of view. The verbal accompaniment to the children’s experiences is what allows them to frame their understanding. One of the things we need to ensure, as teachers, is that children are given a variety of experiences that allow them to engage with fractions as both the names of numbers and also as operators.
Reference
Haylock, D., & Cockburn, A. (1989, January 1). Why do fractions and decimals seem difficult to teach and learn? Retrieved March 4, 2015, from http://www.annery-kiln.eu/gaps-misconceptions/fractions/why-fractions-difficult.html
Group members: Rishmattie Maharaj, Christine Jainth, Angallie Maraj, Maya Dass
ReplyDeleteDifficulties with Fractions
The understanding of the concepts of fractions is very important in understanding equivalent fractions. Students have considerable difficulties with fraction equivalence, the notion that different fractions can represent the same amount (Bana, Farrell, & McIntosh, 1997; Pearn, Stephens, & Lewis, 2003 as cited in Anderson & Wong, 2007, p. 1). Throughout our life, people have known that one number represents one quantity.
Kamii & Clark (1995) reported that it is vital to understand equivalent fractions because it develops a base for understanding addition and subtraction of fractions and allows students to compare and order fractions. Failure in understanding the concept of equivalent fraction in primary school can cause students to face difficulties in higher level education especially in the understanding of algebra.
Reference
Anderson, J., & Wong, M. (2007). Teaching Common Fractions In Primary School: Teachers‟ Reaction to a New Curriculum. Proceedings of the Australian Association for Research in Education’s 36th Annual International Education Research Conference, Volume 1. (pp. 1-13). Retrieved January 29, 2009, from http://www.aare.edu.au/06pap/and06181.pdf
This comment has been removed by the author.
ReplyDeleteGroup 5
ReplyDeleteGroup members Fazida Yathali, Reshma Gangadeen, Ornella Oudai, Abdur Mohammed
The importance of fractions
Students in our primary schools do not necessarily see the importance of factions at their young, tender age. This now becomes a requirement of the teachers to show their students the importance of fraction. Fractions are the pillars for almost every other math topic there is and if the students do not see the connections then it would be very problematic for the child's future in mathematics. Fractions enable students to grasp the nature of numbers and their concepts. When fractions are misinterpreted it will interfere with the ability of the child to quickly understand mathematical processes later on in life such as algebra. This is only looking at an angle from school however we meet fractions everyday in our lives from buying groceries to even cooking food fractions play a very important role for everyone.
Reference
Hanich, L. (2009, July 7). Why are fractions so important? Retrieved March 7, 2015, from http://www.svsd.net/cms/lib5/PA01001234/Centricity/Domain/1/theparentpage/articles2/161.pdf
Teaching and learning maths/fractions at an early level can often be difficult for students to understand primarily because of the the way it is introduced to children.
ReplyDeleteAs many teachers and parents know, learning the various fraction operations can be difficult for many children. It's not the concept of a fraction that is difficult - it is the various operations: addition, subtraction, multiplication, division, comparing, simplifying, etc. of fractions
The simple reason why learning operations can prove difficult for many students is the way they are typically taught. Just look at the amount of rules there are to learn about fractions!
1. Fraction addition - common denominators Add the numerators, and use the common denominator
2. Fraction addition - different denominators First find a common denominator by taking the least common multiple of the denominators. Then convert all the addends to have this common denominator. Then add using the rule number 1.
3. Finding equivalent fractions Multiply both the numerator and denominator by a same number.
4. Convert a mixed number to a fraction Multiply the whole number part by the denominator and add the numerator to get the numerator. Use the common denominator as in the fractional part of the mixed number.
5. Convert an improper fraction to a mixed number Divide the numerator by the denominator to get the whole number part. The remainder will be the numerator of the fractional part. Denominator is the same.
6. Simplifying fractions Find the (greatest) common divisor of the numerator and denominator, and divide both by it.
7. Fraction multiplication Multiply the numerators and the denominators.
8. Fraction division Find the reciprocal of the divisor, and multiply by it.
9. Comparing fractions Convert the fractions so they have a common denominator. Then compare the numerators.
10. Convert fractions to decimals Divide using long division or a calculator.
If students simply try to memorize these rules without knowing where they came from, the rules will probably seem like a meaningless jungle. They probably won't seem to connect with anything about the operation, but instead work like "magic": you multiply, divide, and do various things with the numerators and denominators to come up with the answer.
Students can then become blind followers of the rules, tossing numbers here and there, calculating this and that - and getting answers without having any idea if they are reasonable or not. Besides, it is quite easy to forget these rules or misremember them - especially after 5-10 years.
The solution: manipulatives and visual models
Instead of merely presenting a rule, a better way is to use visual models or manipulatives during the study of fraction arithmetic. That way fractions become something concrete to the student, and not just a number on top of another without a meaning. The student will be able to estimate the answer before calculating, evaluate the reasonableness of the final answer, and perform many of the simplest operations mentally without knowingly applying any "rule."
Now, typical textbooks DO show visual models for fractions, and they DO show one or two examples of how a certain rule connects with a picture. But that is not enough! We need to have children solve lots of problems using either visual models or fraction manipulatives. Another way is to ask them to DRAW fraction pictures for the problems. That way the students will form a mental visual model and can think through the pictures.
Fractions is undoubtedly one of the more challenging topics to teach to young learners. Teachers are often forced to try "different approaches" to teaching fractions because students seem unable to grasp the concepts. Actually, students have a knowledge of the concepts of Fractions, however, problems begin when they are required to understand the operations for working with Fractions.
ReplyDeleteThere is a growing debate regarding why fractions pose so may challenges to young learners. One argument is that students struggle with Fractions because of HOW it was taught to them. This perspective places the blame squarely on the educators; in other words, the reason students struggle with Fractions is because of "us".
Leah rightly referenced a solution to this dilemma, "Instead of merely presenting a rule, a better way is to use visual models or manipulatives during the study of fraction arithmetic." Students need to be able to make sense of what they are learning. If students are mainly presented with a host of rules and formulae, they would not develop a working knowledge or understanding of Fractions. As with any other area of teaching and learning, students must be taken from the concrete to the abstract in order to help develop their understanding of a subject or topic.
In light of this, teachers need to pay closer attention to the previous knowledge of their learners as this will be the foundation on which they build. A child's first introduction to fractions is also of critical importance. It must be done correctly the first time in order to maximize the learning experience for the child. If fractions were not introduced properly, the child would not be successful at it. The child may also develop a mental block towards Fractions because of the initial classroom experience.
Successful strategies for teaching Fractions to young children have been developed. Teachers now have the privilege of being taught the Science of teaching young children Mathematics. While different methods may be presented, two approaches stand out:-
(1) If students are to overcome challenges with Fractions, they must be allowed to
discover the meaning of the relationships.
(2) Students must be allowed to work with relevant concrete or tactile stimuli in
order to make sense of Fractions.
Hence, the word "challenges" with reference to Fractions and children's learning should become increasingly obsolete as teachers now have the tools and knowledge of how to effectively deliver their instructions.
Group members: Rishmattie Maharaj, Christine Jainth, Angallie Maraj, Maya Dass
ReplyDeleteThe Challenge
Fractions are the earliest topic in school mathematics where educators agree that students fail and teachers struggle to teach. “Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra.” (National Math Advisory Panel, 2008.) This challenge is understandable as fractions present major conceptual leaps for students. Consider these factors:
Fractions can describe many different things, including parts of a whole, parts of a set, time, and length.
Sophisticated reasoning is required to evaluate any fraction. Students must analyze the relationship between two numbers in order to understand a single value, and recognize that the real value of a fraction is dependent upon the unit, or whole, of which it is a part.
Fractions operations can be multi-step and abstract. Addition and subtraction can require multiple steps, while multiplication and division are too abstract for many people.
Fractions present a plethora of new terms for students to master. Terminology like numerator, denominator, equivalent, common, uncommon, proper, and improper can confound students, especially students who are struggling with reading.
Our group came up with two solutions to help teachers and students with this problem. A solution to this problem is by using visual models to progress from concrete to abstract. According to the NCTM Standards, educators have consistently agreed on the value of moving from concrete manipulative, to pictorial representations, to abstract numbers. Teachers should start their lessons with the visual models and the language, and then lead students to understanding fractions and solving number problems. Another solution to this challenge was to provide concentrated time with supplemental lessons. Instructional time is a key to success, and every instructional minute is precious. Students can successfully complete as many problems in 30 minutes. Students experience success with instruction that unfolds logically and adapts to their level. The concentration of successful interactions builds synaptic connections and sets the stage for success.
Group #5
ReplyDeleteMembers: Fazida Yathali, Ornella Oudai, Reshma Gangadeen, Abdur Mohammed.
Learning and Teaching Fractions
A fraction is part of a whole. It's less than 1 whole thing, but more than 0. We use fractions all the time in real life. Have you ever ordered a quarter-pound burger? Or noticed that your gas tank is half full? Both of these are fractions of the whole amount, a whole pound of meat, or a whole tank of gas. Every fraction has two parts: a top number and a bottom number. In math terms, these are called the numerator and the denominator. As long as you remember what each number means, you can understand any fraction. The top number, or numerator, refers to a certain number of those parts. It lets us know how much we're talking about. There are many ways how to teach fractions, and the success or failure of the student to understand fractions will depend on the teaching method used. It's been said that if a student understands fractions, then they can understand any mathematics concept. It is then very important for every math teacher to know how to teach fractions in the most approachable way possible. When deciding on a method of how to teaching fractions, we need to use fractional analogies that the student will immediately recognize. Thus enters the pizza as the perfect instrument needed to teach the concept of the fraction. All students learn differently therefore several methods of teaching should be used when teaching fractions.
How to Teach Fractions. (1998). Retrieved March 13, 2015, from http://www.mathgoodies.com/articles/teach_fractions.html
Free Fractions Tutorial at GCFLearnFree. (1998). Retrieved March 13, 2015, from http://www.gcflearnfree.org/fractions
Group members: Rishmattie Maharaj, Christine Jainth, Angallie Maraj, Maya Dass
ReplyDeleteThe Essential Components of Understanding Fractions
The topic of fractions is an integral part of the elementary school mathematics curriculum, yet it is one of the most difficult for students to master. Understanding and being able to use fractions is essential for mathematics success; however, traditional teaching methods often focus on procedures and set of rules rather than deep conceptual understanding that is necessary for fraction success. “Children are bound to find fractions computations arbitrary, confusing and easy to mix up unless they receive help understanding what fractions and fraction operations mean” (Siebert & Gaskin, 2006, p. 394). Teachers feel that teaching fractions is a challenge because they must consider what will help to deepen students’ understanding (Yoshida & Shinmachi, 1999).
In order to assist children in developing a deep conceptual understanding of fractions, teachers must realize that fractions are not simply “algorithms to be taught (Faulkner, 2009, p. 28). Teachers should understand that fractions are in fact a vital part of each component of number sense.
By considering how fractions are directly related to each aspect of number sense, teachers will be better able to understand the important mathematical bases for fractions and be able to use these as a foundation for teaching fractions to students in a way that supports linking fractions to other aspects of mathematics, thus deepening conceptual understanding. In addition to understanding how fractions are integrated into number sense, teachers should use a number of methods for increasing conceptual understanding, such as representations, models, manipulatives, and games; yet make sure that they are developmentally appropriate, relevant to real-life, and do not cause misconceptions.
One area of improvement could be on how teachers engage students. Students should be shown how fractions apply to their personal lives. This could make the lesson more motivating and successful for the students. Many teachers do not make the connections from the content to real life applications. Many of the fraction lessons taught are not engaging or actively involve the students. Instead, they consisted of the teacher providing examples and the students practicing.
References
Brizuela, B.M. (2006). Young children’s notations for fractions. Educational Studies in Mathematics, 62(3), 281-305.
Chick, C., Tierney, C., & Storeygard, J. (2007). Seeing Students' Knowledge of Fractions: