Paraphrase the attached document "maths assignment"
PHYSICAL AND VIRTUAL MANIPULATIVES INTRODUCTION In the subject mathematics, a manipulative is seen as an object that is designed for learners to distinguish mathematical concepts by manipulating it. By the use of manipulatives, learners are able to learn concepts through developmentally appropriate experiences. Through this essay I will look at two manipulatives being physical and virtual manipulatives, there will be a definition of both manipulatives with similarities and differences, a discussion on the importance of manipulatives with links to theories of learning, and evaluation of the effectiveness of both manipulatives in the teaching and learning of maths, at this point I will look at principles of Piaget, Bruner and Skemp and how the progression of the concept is developed through the use of manipulatives. Lastly, there will be two activities which includes, a physical manipulative and a virtual manipulative. DEFINITION, SIMILARITIES AND DIFFERENCES OF PHYSICAL AND VIRTUAL MANIPULATIVES Physical manipulatives are tangible objects of which learners use to discover mathematical concepts through their visual and tactile senses. Objects of physical manipulatives include everything from lego’s, rods, blocks and magnets. Which allows for learners to engage with various mathematical concepts. Virtual manipulatives are interactive, web-based visual representations of dynamic objects that presents chances for constructing mathematical knowledge (Mayer, Bolyard and Spikell 2002:373). Virtual manipulatives include any digital objects, such as a tablet, computer, easi teach, calculator and any internet-based resources. A similarity among the two manipulatives are that, digital objects resemble physical objects and can be manipulated by the use of a mouse in the same ways you would with real objects. These two manipulatives have many differences among them. When looking at the differences among the two manipulatives, we look at aspects such as the meaning of the manipulatives, the centre of learning, the style of learning, the century of learning, cost factors, actual material being used to learn and examples of materials being used to learn. When looking at the meaning of the manipulatives, physical manipulatives state that, real objects are being used, therefore, it allows for information to be visually and kinaesthetically received (Hunt, Nipper and Nash 2011:6) as well as, it is simpler and objects are much more moveable. Whereas virtual manipulatives mean, using technological objects to teach and learn mathematical concepts, therefore, representations regarding maths are manipulated and displayed on a computer or tablet. The next aspect being looked at is, the centre of learning, physical manipulative material is seen as teacher centred, the educator is able to explain misunderstandings and is able to form connections among mathematical concepts and illustrations and will inspire more detailed and richer understanding. With virtual manipulative material, it is more learner centred, by using a computer or tablet different illustrations of the same maths concept is given to learners but they have to figure out on their own which concept best suits them, in other words, the learner will guide their own learning. When looking at the style of learning, when using physical manipulatives, the material used to learn a specific concept is provided by the educator. Therefore, by using materials chosen by the educator it makes it more helpful for learners to understand the concept being taught, as learners will be able to work with these materials step by step and will be able to see what they are doing. Whereas virtual manipulatives, learners have to explore different methods of solving a concept on their own. They will have to explore different apps to find a suitable method to solve a concept which in turn, will take up more time and the child will feel rushed to complete tasks (Hunt, Nipper and Nash 2011:6). The next aspect covers the century of learning, the physical material leans more on the traditional way of teaching, older methods of teaching is used. Concepts are broken down for learners in a way that they will never forget. With virtual manipulatives, it leans more on the modern way of teaching, it grabs the attention of the more technological generation. When looking at the cost factor, physical manipulatives are more costly as they require one to purchase materials to make resources or to purchase resources needed, whereas with virtual manipulatives there is hardly no cost as everything is done online using different apps, but due to many South African schools being underprivileged it would be cheaper to use physical manipulatives as appose to virtual, as many people cannot afford technological devices. Finally, we look at the aspects, materials used, and examples of material being used to learn. With physical material, learners are able to make use of their senses such as touch and see, physical material allows for learners and educators to be more selective with choosing pieces to work with. Virtual material, learners are not able to touch objects, but they can see it on a screen. Examples of physical materials are pencils, sticks and rods, so learners work with them and it develops their kinaesthetic ability. With virtual material, only apps are used so learners do not get the opportunity to be hands on with their learning. IMPORTANCE OF MANIPULATIVES WITH INSIGHTFUL LINKS TO RELEVANT THEORIES OF LEARNING When looking at the performance of South African learners in the area of mathematics, it is looked at as big concern, according to the Western Cape Education Department from the systemic maths assessment in grade 3. The result on this research revealed that less than half of the learners in South Africa attained a 50% pass rate in maths between the years 2010 and 2013(Siyepu 2013:1). By these findings it proposes that teaching approaches being used are not producing adequate results in South Africa. Therefore, the use of manipulatives when teaching maths concepts is vital, as maths manipulatives help make abstract ideas concrete, as a picture is worth a thousand words, learners will be able to use manipulatives to construct physical models of abstract mathematical ideas. Manipulatives lift maths of textbooks; learners are able to connect mathematical vocabulary and symbols and construct an understanding of ideas. Manipulatives enhance the learning of maths and makes it more interesting and enjoyable, as educators are able to be creative and use different resources and methods to intrigue and motivate learners to learn maths. Manipulatives builds confidence in learners and allows for trial and error to take place, if learners see physical evidence of the manner in which they are thinking, the child’s understanding becomes more robust (Jones 2015:01). Manipulative material serves as concrete models for learners when conducting problem solving, therefore, manipulatives are of importance as it is a useful tool for solving problems. In the 21st century, learner’s ability of learning differs, by the use of different manipulatives teachers are able to modify and adjust learning to suit a child’s level of absorbing knowledge. The use of manipulatives to teach, positively affects learning, learners grasp concepts with great ease with the incorporation of manipulatives in maths lessons making lessons effective. EFFECTIVENESS OF EACH TYPE OF MANIPULATIVE IN THE TEACHING AND LEARNING OF MATHEMATICS When looking at the theoretical basis of the importance of the use of manipulatives within the context of mathematics the use of manipulatives is not new. Piaget’s theoretical work supports the use of manipulatives, according to his theory, learners go through four stages of intellectual development being sensorimotor (birth- 24 months), preoperational stage (24 months- 7 years), concrete stage (7-12 years) and formal operational (adolescence through adulthood). We look at only two stages being the concrete and formal operational as it relates to the foundation phase learners. In concrete operations stage, learners utilize their senses in order to understand concepts (Ojose 2008: 27). It is at this stage that learners learn effectively through several illustrations and experience concrete materials while being hands-on. By learners being hands on, learners’ confidence is built, and it allows for learners to see physical evidence of their thinking, they have a better understanding of problem solving and by being hands-on learning becomes enjoyable and fun during this stage. When learners progress to the formal operations stage, the need for concrete experiences diminishes, but never ceases. This suggests that manipulatives hold a value in the lower secondary classroom and is a time when learners are making the transition from concrete to formal operations. The importance of the use of manipulatives also forms a theoretical basis around Harwood Gardner’s theory of intelligences (Duchesne and McMaugh2016:5), according to this theory there are eight separate domains of intelligences that allows for learners to work best from. By using manipulatives, it carries in essentials of visual, auditory, kinaesthetic and tactile learning, which allows for a wider range of learning styles to be experienced. When looking at Vygotsky’s sociocultural theory, we become aware of learners learning through the zone of proximal development (ZPD). Vygotsky’s theory is founded on the fact that ideas of knowledge is represented as eternal construction, reconstruction and deconstruction of reality, this will vary according to the practices of each learner. According to Vygotsky, a child’s individual and learning development is reliant on their socialization with their environment (Hall 2007:94). Therefore, interactive learning as well as informal activities are vital as grownups offer learners new information, support their skills progression and extends their abstract understanding (Ramani and Siegler 2014:2). The ZPD is seen as the distance between the actual growth level, which is reliant on self-governing problem solving, the level of possible growth is determined through problem solving, which is done through guidance of adults or a capable peer. Through Vygotsky’s theory, the importance of manipulatives provides learners with the opportunity to learn maths through interaction with knowledgeable others, they provide the necessary support and tools to scaffold the process of learning in the ZPD. The ZPD is divided into four stages and attempts to explain what happens at the beginning of teaching a concept, during and after the teaching. In stage 1, a knowledgeable other (educator) comes up with different teaching interventions which include modelling, coaching and other methods of scaffolding to assist with teaching a concept. The teacher uses appropriate manipulatives to assist learners with learning maths concepts. Stage 2, the learner tries to make sense of a concept and carries it out independently with assistance of the manipulative given, stage 3, the child will now be able to perform solving a maths problem on their own. The learner is now able to grasp a concept without assistance from the educator. Finally, in stage 4, the de-automatization of performance moving back over to the ZPD and starting from phase 1 again. Educators need to be mindful of different levels of learning and that different manipulatives work best with different learners. LINKS BETWEEN THEORY AND PRACTICE When looking at Piaget’s theory, we look at four basic principles of his theory, being schemes which organize experiences, assimilation meaning incorporating new experiences into existing schemes, accommodation which means changing schemes based on experiences and equilibration meaning recognizing