To effectively teach place value (PV), teachers must understand why it is important and how children best learn about it. This requires teachers to have pedagogical content knowledge (PCK) of PV to effectively teach and assess children on their mathematical content knowledge (MCK) of PV. Educators must be knowledgeable of the stages that PV needs to develop, and the important things children need to understand about PV. Teachers must be familiar with the typical experiences needed to help children learn PV concepts, how specific ideas about PV develop from earlier ideas and how they underpin what children learn later. Teachers should use common resources, manipulatives, and equipment to help children develop a conceptual understanding …show more content…
According to Booker et al. (2014) it is necessary when learning about PV that children can relate what they are learning to real-life situations. This view is supported by Van de Walle (2006) who states children should be presented with authentic activities focusing on the three main components of PV: grouping activities, giving oral names for numbers and written symbols for the concepts. Further to this, in a study by Ross (2002) he noted children improve their ability to express mathematical thinking when they have been given opportunities to discuss and write down ideas. Therefore, through a wide range of authentic experiences, activities and opportunities children will be able to visualise and conceptualise PV concepts. As PV develops from earlier mathematical ideas it underpins what children learn later. Skemp (1989) states PV is a secondary concept and to learn PV it is essential children understand primary mathematical concepts. According to Gunningham (2011) the key skills and understandings required to understand PV are counting, subitising, part-part-whole, trusting the count and composite units. This view is supported by Booker et al. (2014) who asserts to understand PV children must first visualise numbers, then use materials to talk about them and understand concepts such as zero and …show more content…
According to Booker et al., (2014) typical difficulties children experience as they develop their understanding result from misconceptions or gaps in understanding. They also state children often confuse similar sounding names, write numbers in the wrong order and have difficulty comparing numbers. It is vital, according to Booker et al., (2014), to overcome these difficulties and misconceptions, that teachers follow a specific sequence of steps to establish number understanding because when children meet ‘powerful ideas’ for the first time they must be presented in accord with their needs (Booker et al., 2014). Three of the most common confusions or misunderstandings are the confusion of teen numbers, misinterpreting specific vocabulary and confusion relating to the concept of zero. Therefore, to overcome difficulties and misconceptions held by children, teachers must assess students regularly to ascertain if there are any gaps in understanding before moving to the next
Piaget believed that children in this stage experience two kinds of phenomena: pretend play and Egocentrism. Pretend play is the ability to perform mental operations using symbols. Egocentrism is the inability to perceive things from a different point of view. For example, a child covering his own eyes, because he believes that if he can’t see someone, then they can’t see him as well. When a child is seven to eleven years old, it is in the concrete operational stage. At this point, Piaget believed that children are able to grasp the concept of conservation. Conservation is the principle that mass and volume remain the same despite the change in forms of objects. For example, children at this age are mentally capable of pouring a liquid in different types of containers. Piaget also believed that at this age a child is capable of understanding different mathematical transformations. At the age of 12, children reach the Formal Operation stage, the final stage in Piaget’s stages of Cognitive Development. This is the
The students are building on the knowledge they have gather from their daily routine, and are able to more easily grasp the concept of place value. Furthermore, in Piaget theory “children need many objects to explore so that they can later incorporate these into their symbolic thinking” (Gordon, & Browne, 2010, p. 106). The teacher gave students a variety of materials to understand the lesson in a pleasurable way. Equally important, the students’ scheduling was derived from Piaget as the students had “plenty of time to explore” (Gordon, & Browne, 2010, p.
Siegel, L. (1982). The development of quantiy concepts: Perceptual and linguistic factors. Children's logical and mathematical cognition , 123-155.
In the Variables and Patterns of Change (Annenberg Media, 2004), we are introduced to two classrooms during their first week of instruction. The first class is Ms. Green’s algebra. Ms.Green uses real life situation of wanting to get a pool in her backyard to teach dimensions and equations. During the example, she helps to guide the students learning by asking leading questions to help them figure out the problem. Once they understand the problem, she puts them into groups to figure out dimensions of different pool sizes and how many tiles it would surround them. While in groups, Ms. Green goes to each group to check their progress and answer any question.
One of the most influential of Piaget’s findings was his theory on the developmental stages of children’s cognition. The developmental stages consist of periods of months or years in which development occurs (Ojose, 2008). There are four Stages of Cognitive Development, the sensorimotor stage, the pre-operational stage, the concrete operational stage, and the formal operations stage. This theory has shaped the way many educators have shaped their lesson plans for years, for example, According to Bobby Ojose (2008), Piaget’s “work on children’s quantitative development has provided mathematic educators with crucial insights into how children learn mathematical concepts and ideas” (p. 26).
...things together. Therefore, arithmetic and books that teaches logic are introduced to a child at this stage. For example, a child is taught basic addition and subtraction, that is one plus one, two, three and so forth. In so doing, a child develops skills to make simple decisions and judgment. Their skill of reasoning is also enhanced. Thereafter, a child grows to the normal school ongoing age. Here, such children have to be taught to internalize with the environment in a more effective way. They mental capacity is much greater to accommodate more aspects of reasoning and logic. Teachers use books such as story books, advanced mathematics integrated with social interaction so that they discover things by their own. The main objective is to get them effectively interact with the environment. This enhances their development towards normal functioning human beings.
Analyzing and describing: the children begin to analyze shapes and their properties (sides, edges, angles, etc.).
“Cognitive development refers to how a person perceives, thinks, and gains understanding of his or her world. Cognitive development is the construction of thought processes, including remembering, problem solving, and decision-making, from childhood through adolescence to adulthood.”(Cog.) J is in Jean Piaget’s preoperational stage of development because he is still learning how to conserve. I observed this when I completed my math interview with J. When I showed him two clay balls he agreed that they were the same size, but when I flattened one of the balls he believed they had a different mass because they looked different. He shows some signs of being close to understanding of conservation of numbers. I set up two equally spaced rows of counter and J told me that the two rows were the same. Then I spread out one row to look longer. At first, J said that the longer row contained more counter, but when I asked him how he come to that conclusion he counted each row and realized that they had the same amount. J was able to show me his thought process and solve the problem by applying math skills that he already
N.G., 4 years, 11 months, embodied all I could ask for in a child to conduct such an interview on. Nearing her fifth birthday in the upcoming week, her age is central between ages three and seven, providing me with information that is certainly conducive to our study. Within moments upon entry into our interview it was apparent that my child fell into the preoperational stage of Piaget’s cognitive development. More specifically, N.G. fell into the second half of the preoperational stage. What initially tipped me off was her first response to my conduction of the conservation of length demonstration. Upon laying out two identical straws, her rational for why one straw was longer than the other was, “it’s not to the one’s bottom”. This is a perfect example of an intuitive guess, though showing a lack of logic in the statement. A crucial factor of the preoperational stage of development is that children cannot yet manipulate and transform information into logical ways which was plainly seen through the conservation of number demonstration. Though N.G. was able to correctly identify that each row still contained an equal number of pennies upon being spread out, it required her to count the number of pennies in each row. In the preoperational stage of development children do not yet understand logical mental operations such as mental math as presented in the demonstration. Another essential element that leads me to firmly support N.G.’s involvement in the preoperational ...
Children tend to learn more when they know why what they are learning is important and if that material is presented in an interesting way. Take for example a preschool teacher who needs to teach her class about the different shapes. Instead of just showing her students a poster with the different shapes on it, she has her students get out of their seats and begin exploring the classroom for differently shaped objects. Once they have found some objects, have them share with the class what the object is and what shape it is. This exercise will not only help the other students learn the shape of the object, but it will also help the student who is sharing. Piaget believed that children’s cognitive growth is fostered when they are physically able to experience certain situations. By having students share with the class what shapes their objects are, they are fostering their public speaking
The more common notion of numeracy, or mathematics in daily living, I believe, is based on what we can relate to, e.g. the number of toasts for five children; or calculating discounts, sum of purchase or change in grocery shopping. With this perspective, many develop a fragmented notion that numeracy only involves basic mathematics; hence, mathematics is not wholly inclusive. However, I would like to argue here that such notion is incomplete, and should be amended, and that numeracy is inclusive of mathematics, which sits well with the mathematical knowledge requirement of Goos’
According to Piaget in the “preoperational stage, which goes through 2 to 7 years of age a child should have the ability to use symbols to represent objects in the world and thinking remains egocentric and centered” (Slavin ,2015) For example, I lined up two sets of quarters on a table in front of Ahmad. Each set of quarters had four in a row, I asked Ahmad which set of quarters had the most he told me that they all had the same amount. For the second part I lined the quarters up differently, but they still had the same amount the second row of quarters I spaced them out. I then proceeded to ask Ahmad the same question which row of quarters had the most he replied the second row. I asked Ahmad why did he think the second row had the most, he replied because it is larger. This method would be conforming to Piaget’s principle of conservation, “one manifestation of a general trend from a perceptual-intuitive to an orientation, which characterizes the development of conceptual thinking” (operational Zimiles
This representation is called preverbal number knowledge, which occurs during infancy. Preverbal number knowledge occurs when children begin representing numbers without instruction. For instance, children may be familiar with one or two object groupings, but as they learn strategies, such as counting they can work with even larger numbers. As stated in Socioeconomic Variation, Number Competence, and Mathematics Learning Difficulties in Young Children “Thus only when children learn the count list and the cardinal meanings of the count words, are they able to represent numbers larger than four” (Jordan & Levine 2009, pp.61). Typical development occurs along a continuum where children develop numerical sense, represent numbers and then begin to understand the value of the numbers. These components are required when differentiating numbers and
Mathematics teachers teach their students a wide range of content strands – geometry, algebra, statistics, and trigonometry – while also teaching their students mathematical skills – logical thinking, formal process, numerical reasoning, and problem solving. In teaching my students, I need to aspire to Skemp’s (1976) description of a “relational understanding” of mathematics (p. 4). Skemp describes two types of understanding: relational understanding and instrumental understanding. In an instrumental understanding, students know how to follow steps and sequential procedures without a true understanding of the mathematical reasons for the processe...
Skemp, R. (2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12 (2), 88-95. (Originally published in Mathematics Teaching, 77, 20-26, 1976)