Children's Mathematical Development - Essay Example

Task 2c Identify and explain three specific problems that children have encountered, or may encounter, during the development of mathematical or numerical understanding. Children in the early childhood stage encounter many difficulties in learning beginner’s math skills and concepts. Three of their common challenges include the recognition of numbers, knowledge of quantity and one-to-one correspondence. These three problems are illustrated in the following examples and how they were supported by their teachers in understanding math concepts. 1. Child: Nathan Age: 3 years Specific Problem: Number recognition Expected level of skill for age: Differentiating shapes from other shapes; Visual discrimination of numbers (at least 1 to 3) Case: Three year old Nathan would proudly count from 1 to 10 by rote. However, he does not seem to associate the numbers he recites with the symbols that represent the numbers. His teacher has previously assessed that he knew his shapes, namely: circle, square, triangle, rectangle, oval and heart. This shows that he can visually discriminate one object from another. Nathan can not do the same with numbers. He cannot even recognize which are numbers and which are letters. For him they just look like squiggles and strange marks. Support: Introducing numbers to children should expose them to the visual appearance of each number, the quantity it represents and even the strokes used when writing it down. Flashcards of numbers may be used to imbed the appearance of the number to the child’s mind while its name is repeatedly said. An effective method is teaching counting songs and when the number is mentioned, up comes the flashcard of that number. Children love singing and movement and incorporating these in their learning a concept becomes interesting and fun for them. A story that focuses on a particular number (ex: 2 friends found 2 birds eating 2 apples) is likewise interesting to listen to. A teacher may provide a coloring sheet with a big print of the specific number discussed. These methods may be used with Nathan. The strategies address various learning styles that may suit any child. The variety of activities address the needs of the visual (flashcards and story pictures), auditory (songs and story), kinesthetic (movements to songs) and tactile (coloring sheet) learners. Resources/ Materials used: number flashcards; CD’s of number songs (Five Little Monkeys; Five Green Speckled Frogs; There were 10 in the Bed.. etc.); number storybooks; number coloring sheet focusing on 1 number at a time, box of crayons. 2. Child: Hannah Age: 4 years Specific Problem: determining quantities of numbers Expected level of skill for age: number identification up to 10; knowledge of quantities up to 10 matching numbers to quantities Case: Hannah has mastered identifying numbers up to 10 and visually comparing sets with more objects or less objects if the quantities are obvious (like comparing a set of 2 buttons with a set of 10 buttons). However, when asked to match a number with its quantity, she wouldn’t know what to do. She has mastered identifying the numbers but would not have a clue what it represents in quantity. Support: The teacher needs to provide concrete materials for Hannah to count such as buttons, candies, blocks or even her classmates. Number cards must be available at hand so when Hannah learns to count the concrete object and knows that the last number counted is the quantity of the set, then she looks for the correct number card to match the set to. The teacher needs to give her 2 options of numbers at a time, and as Hannah gains more skills in quantities, then she can also increase the number of options of number cards to choose from. Later on, when she is adept at counting concrete objects, she may be given a worksheet with pictures of sets of objects that need to be matched with its correct number. Resources/ Materials used: concrete objects for counting; number cards; worksheet for matching sets with its numbers; box of crayons. 3. Child: Billy Age: 4 years Specific Problem: one-to-one correspondence of objects Expected level of skill for age: number identification up to 10; knowledge of quantities up to 10 matching numbers to quantities Case: Like Hannah, Billy can rote-count objects and identify numbers, but lack understanding of one-to-one correspondence when matching objects from different sets. He needs to learn that for a corresponding object, there is another object to match it. For instance, for one hand, there should be one glove. Support: Billy needs to have a lot of practice in matching corresponding pairs of things. Then, his teacher can assign him practical tasks to evaluate his understanding of the concept of one-to-one correspondence. One task is asking him to count the number of classmates he has and set the table with a place for each child, including himself. Each place in the table should have one chair, one placemat, one bowl and one spoon for each child. He can also have similar activities as matching a set of bottles with its bottlecaps, so he learns that one bottle needs one bottlecap and not two. These tasks will be useful when he works on activity sheets which require him to match objects from one set with objects from another set by aligning each pair with a line to determine if they are equal sets or if one set has more or less objects than the other set. Resources/ Materials: table settings that include tables, chairs, placemats, bowls and spoons depending on the number of children in class; set of 5 bottles and 5 bottlecaps; activity sheet of matching objects from one set to objects from another set by aligning each object with a line; crayon. Task 3: In this section include four Activity Plans detailing numeracy activities you have carried out with children. Clear, specific and referenced learning intentions/ objectives must be detailed for each activity, taking full account of current initiatives of the Early Years Foundation Stage. Task 3b would be two of the four activity plans that are to incorporate the use of ICT. The evaluation sections of the Activity Plans must be completed in order to demonstrate your ability to evaluate and analyse planning and preparation, the role of the adult and childrens learning. 3 A. Hands-On Numeracy Activities EYFS recognizes the importance of play in childhood, as it provides many benefits. Play is a child’s world. It is an activity where one can be free to be oneself without anyone imposing rules or restrictions. Play offers many benefits that foster children’s learning. It engages the mind to actively imagine various scenarios for fun or for problem- solving. Practitioners should have enough flexibility in planning activities for children. Following their lead in terms of interests shared by the majority of children is one effective way of capturing their attention and motivating them to develop skills. Usually for children in the EYFS stage, it usually involves play-like activities. 1. Fishing for Numbers: a. Learning Objective: i. To teach preschool-aged children identification of numbers ii. To practice eye-hand coordination iii. To develop persistence and patience iv. To develop concentration v. To develop confidence and enhance self-esteem b. Procedure: i. Prepare cardboard cut-outs of fish in various plain colors. Write a number in each fish (1-10 only). Insert each fish in a paper clip each. Put all the cut-out fish in a large basin on the floor. ii. Give each child a long “fishing rod” made out of short sticks (about 1.5 ft. long) with a yarn tied to one end and a piece of magnet hanging from it. iii. Let each child “fish” for a number fish by trying to attach the hanging magnet to a paper clip and letting the fish hang from it without dropping. iv. Let the child read the number on the fish he has successfully “caught” and keep it if he is correct. If not, he should throw the fish back to the basin until he catches one he correctly identifies. v. Children can sing the following song before and after fishing just to strengthen the concept of the activity. “1 2, 3, 4, 5, Once I caught a fish alive. 6, 7, 8, 9, 10, then I let it go again. Why did you let it go? Coz it bit my finger so Which finger did it bite? This little finger on my right.” c. Evaluation: Teacher evaluates if children can identify the numbers or not. Apart from this numerical competency, the child’s attitude and behaviour during the game are also evaluated, as the activity is meant to develop other socio-emotional values and skills. Teacher can also assess how astute his eye-hand coordination is in trying to direct the fishing rod to a certain fish on the basin. 2. Pinning Balls: a. Learning Objective: i. To help preschool children give the quantity asked for by the number. ii. To develop fine motor coordination iii. To develop concentration. b. Procedure: i. Give each child a set of cardboard cut-outs of balls. In each ball, a number should be written (1-10 only). Give also a box of clothespins. ii. Let each child pin the same number of clothespins around the cardboard ball as the number indicated on each ball. iii. Let child do the same for all the balls in his set. c. Evaluation: Children who have gained understanding of the concept of quantities of numbers will have put the correct number of clothespins on each ball. The teacher can also assess if child had difficulty in manipulating the clothespins to attach to the cardboard balls. 3 B. Activities Involving the use of ICT 3. Using Math software such as Sesame St., Fisher-Price, Jump Start Kindergarten, etc. a. Learning Objective: i. To expose children to an alternative form of learning which is with computer use. ii. To teach children to follow directions by listening well to the commands or cues in the software used. iii. To teach children basic math concepts such as matching numbers to quantity or a shape to an object with the same shape. iv. To develop eye-hand coordination by dragging the computer mouse or trackball in accordance to movement on the screen monitor. v. To teach children social skills such as waiting for one’s turn and giving chance to others. b. Procedure: ix. Select an appropriate Math computer game for a small group of three-year old children. Choose a game that lets them make choices for answers to questions by dragging the object to a goal. An example is selecting the correct number for the set of objects counted and dragging the number to the set. If the child correctly responds, the computer will give him positive reinforcement by splashing colors and happy sounds on the screen. If not, the computer will ask him to try again. ii. The children will work in small groups (3 per group) and have their alternate turns with the computer. They can answer the questions as a group while the one whose turn it is will manipulate the mouse or trackball. c. Evaluation: In the beginning, the children may have a difficult time following directions, as they are still adjusting to the computer use and to the novelty of the software. As they get the directions, the teacher can evaluate if they can do the appropriate matchings between numbers to quantities or objects to shapes. Behaviour of the children may also be observed if they are able to wait for their turn and if they can listen and follow directions. 4. Teaching Number Strokes Using the MS Paint Program a. Learning Objective: i. To expose children to an alternative form of learning which is with computer use. ii. To teach children to follow directions by listening well to the teacher and to observe carefully what she does on the screen. iii. To teach children writing strokes of numbers on a more colorful, animated and novel medium. iv. To develop eye-hand coordination by dragging the computer mouse or trackball in accordance to movement on the screen monitor. b. Procedure: i. This time, the teacher directly tutors each child on the computer using the program MS Paint. An attachment via the USB of a writing pad and a writing stylus are used to control the cursor on the screen. ii. The teacher shows the child how to write the number one by singing while writing: “We start from the top then we go down (3x) … to write the number one”. The strokes she writes on the writing pad with her stylus will show up on the screen. iii. The child will have his turn writing the number while singing the song too. iv. When the child has mastered writing number one, the teacher can proceed to number 2: “Halfway round then we go out (3x).. to write the number two”… v. This process goes on until they reach the number 10. It does not have to all be done in one session. The numbers can be taught in multiple one-on-one sessions. vi. The child will need to practice writing on paper and crayons too outside their ICT writing tutorial sessions. c. Evaluation: Using the writing pad and stylus on the computer is another novel way to encourage writing in young children. Enhancing the learning experience with a song greatly helps the child retain the directions because of the tune and the repetition of the words in the song. The teacher evaluates if the child has learned the strokes and if he can do them without her assistance. Task 3c: Explain and justify the learning and teaching approaches embedded within your Activity Plans. Identify relevant theories and ideas on how numeracy can be taught and indicate how those theories were reflected in your practice (ie, in the numeracy activities carried out with the children). In addition you should draw conclusions and make recommendations as to how mathematics should be taught in the early years. The Early Years Foundation Stage (EYFS) is a resource for early childhood care and education practitioners to support the needs of young children under their care. It sets standards for learning, development and care for children up to five years of age. EYFS provides a wide variety of information on child development to help practitioners understand how children grow and what they need to help them optimize their potentials. The use of this resource will effectively enable early childhood settings to meet the key outcomes outlined in Every Child Matters and to ensure that high quality service is provided to the children. The specific areas of learning and development, as identified in the Early Years Foundation Stage (EYFS) are: Personal, social and emotional development; Communication, language and literacy; Problem-solving, reasoning and numeracy; Knowledge and understanding of the world; Physical development and Creative development (EYFS, 2007) These are all linked together, as in development in one area affects the others. The practitioner needs careful planning and implementation of activities so that children under their care grow in all areas. The activities discussed above show that it has elements that interlink the different areas of learning and development, and not just focused on learning a math concept. Children in the EYFS state belong to the Pre-Operational period (two to seven years) of Piaget’s Stages of Cognitive Development. This period marks the time when a child becomes able to represent objects and knowledge through imitation, symbolic play, drawing, mental images and spoken language. Lack of conservation skills is also characteristic of this stage. “Conservation is defined as the knowledge that the number, mass, area, length, weight, and volume of objects are not changed by physically rearranging the objects.” (Brewer, 2001, p.318). That is why it is important to always give concrete materials to young children when teaching a math concept since that is how they understand things better. They need to be able to see things concretely first before they can be translated to abstract thinking. Mathematics is a hierarchical discipline where concepts build on previous concepts and more often than not, need full understanding before proceeding to the next, more complicated concept. (Ruthven, 1987). One cannot just jump and teach multiplication without the student understanding the concept of addition. Vygotsky came up with the concept of the zone of proximal development (ZPD). He defined the ZPD as the distance between a child’s independent problem-solving level and that obtained under adult guidance or in collaboration with more capable peers (Wertsch, 1985). Wells (1997) cautioned us, however, that a ZPD is formed not just within an individual learner, but in the interaction between the learner, coparticipants, and available tools during involvement in a common activity. ZPDs, therefore, depend on the quality of the total interactive context as well as individual learner capabilities. (Bonk & Cunningham, 1998). Children usually find it more challenging to learn a slightly more difficult concept to test their mettle in the skills they have gained. “Such cognitive apprenticeships are, of course, inherently reliant on a mentor or guide who effectively uses “scaffolded instruction.” (Bonk & Cunningham, 1998 p.40). As the term implies, scaffolds are temporary supports in the process of learning which are gradually taken away when the student is already capable of learning without them. As an example, the teacher helping the children to do mathematical operations first give them more concrete materials such as paper shapes or beads and as they master the concept, the materials or “scaffolds” are slowly eased away until they can do the operations mentally. This is also done because students may be at a stage when attention span could be short and supports become necessary to hold the children’s attention long enough for the teacher to introduce mathematical concepts. It is essential to strike a balance between giving the pupils sufficient challenge and taking care not to push them into a level they are not yet capable of. Research shows that when children are trained to learn mathematics above their reasoning level, there may be positive results at first but they are “rarely retained unless the child is already in transition from one level to another” (Suydam and Weaver, 1975, p. 47). The teacher should be discerning enough to know when to apply ZPD with her students and know the proper scaffolds to use. In the cases discussed in the firs part, the baseline skills of the children were identified for the teacher to be able to design an activity within the child’s ZPD. According to the behaviorist view, an individual is reinforced (positively or negatively) for responses to various stimuli, hence, the external environment plays a great part in the formation of behaviors. By administering positive reinforcement such as praising or smiling when a desired behavior occurs and administering negative reinforcement such as scolding or correcting when an undesired behavior occurs, one is assumed to encourage the desired behavior and make it more likely that that behavior will recur (Lindfors, 1987). Positive reinforcement works well in bringing out the best in pupils. More important than the lessons taught in the sessions are the interests of the children themselves. This is especially true with very young children whose minds are always brimming with ideas. Being very young, the children must be allowed to express themselves freely. Trafton suggests that individualization must include “acceptance of each child as an individual worthy of adult respect,” and that to this should be added “an acceptance of the child’s ideas, a provision of opportunities for pupil input in developing and selecting learning experiences, a concern for the quality of the child’s intellectual development, and a willingness to take time to know the child as an individual” (1975, p. 39). The activities planned may easily adjusted to accommodate children’s ideas, if any. Mathematics is part of everyday life. Children see it in numbers, counting, in telling time, in measurements, in ordering of sequences, etc. Very young children will benefit greatly if both the home and their school settings will support their mathematical explorations. Teachers and parents may share simple tips or activities with the children to encourage mathematical learning. Parents may learn play-like approaches to assigning home chores to children such as letting them count place settings and set the table according to the number of people who will have dinner. They may also involve children in cooking or baking and letting them follow the recipe and add the correct measurement of ingredients, exposing them to units such as 1 cup, ½ cup, ¾ tablespoon, etc. In doing practical and fun activities like these, the children get to view mathematics in a more positive light that they become more open to learning more about it. References Bonk, C.J. & Cunningham, D.J. (1998) “Searching for Learner-Centered, Constructivist, and Sociocultural Components of Collaborative Educational Learning Tools” in Electronic Collaborators. Retrieved on February 20, 2008 from: www.publicationshare.com/docs/Bon02.pdf Brewer, J., (2001) Introduction to Early Childhood Education, Allyn & Bacon Lindfors, J.W., (1987) Children’s Language and Learning, 2nd Ed. Prentice Hall, Inc. Ruthven, K. (1987). Ability stereotyping in mathematics. Educational Studies in Mathematics, 18(3), 243–253. Suydam, M. & Weaver, F. (1975) Research on learning mathematics. In J. Payme (Ed.) Mathematics Learning in Early Childhood. Reston, VA: The National Council of Teachers of Mathematics. The Early Years Foundation Stage, (2007) Effective practice: Play and Exploration © Crown Trafton, P. (1975) The Curriculum. Mathematics Learning in Early Childhood Education. Reston, VA: The National Council of Teachers of Mathematics. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press. Wells, G. (1997). “The zone of proximal development and its implications for learning and teaching.” Retrieved on February 20, 2008 from http://www.iose.utoronto.ca/~gwells/zpd.discussion.txt Wertsch, J. V. (1985). Vygotsky and the social formation of the mind. Cambridge, MA: Harvard University Press. Read More