The proposal of using the number line to improve number sense for children from 5 to 6 years old in Vietnam

Abstract. Many early theories of learning and cognitive development asserted that young children are devoid of mathematical knowledge before entry into elementary school. The child’s ability in this stage might be enhanced if he is allowed ample opportunity to act on the environment in unrestricted ways in order to start building concepts. Educators of children in this stage of development should lay a solid mathematical foundation by providing activities that incorporate counting and thus enhance children’s conceptual development of number. Developing the number sense of 5 to 6 year-old children is really necessary and important. Thus, to teach numbers and numerical concepts effectively, teachers should use a tool as symbolization shows the relationship between them.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1075.2016-0212 Educational Sci., 2016, Vol. 61, No. 11, pp. 21-28 This paper is available online at THE PROPOSAL OF USING THE NUMBER LINE TO IMPROVE NUMBER SENSE FOR CHILDREN FROM 5 TO 6 YEARS OLD IN VIETNAM Chu Cam Tho1, Nguyen Thi Quynh Anh2 1Hanoi National University of Education 2Vietnam Education Publishing House Abstract. Many early theories of learning and cognitive development asserted that young children are devoid of mathematical knowledge before entry into elementary school. The child’s ability in this stage might be enhanced if he is allowed ample opportunity to act on the environment in unrestricted ways in order to start building concepts. Educators of children in this stage of development should lay a solid mathematical foundation by providing activities that incorporate counting and thus enhance children’s conceptual development of number. Developing the number sense of 5 to 6 year-old children is really necessary and important. Thus, to teach numbers and numerical concepts effectively, teachers should use a tool as symbolization shows the relationship between them. Keywords: Number sense, number line, numerical concept, number and counting, children from 5 to 6 years. 1. Introduction 1.1. The feature of children aged 5 – 6 mathematics cognitive development Children aged 5 – 6 years old in Vietnam as well as others countries are in preschool. This is also the important period of cognitive development of each child. For many years ago, most of parents have children aged 5 – 6 in Vietnam didn’t take care of providing their children with mathematical knowledge especially developing number sense early. Many early theories of learning and cognitive development asserted that young children are devoid of mathematical knowledge before entry into elementary school (e.g., Bereiter & Engleman, 1966; Clapp, 1924; Knight & Behrens, 1928) [7]. However, recent studies and the advent of Piagetian theories about cognitive development have shown the inverse. Piaget’s work on children’s quantitative development has provided mathematics educators with crucial insights into how children learn mathematical concepts and ideas. According to Ojose (2008), Piaget has identified four primary stages of development: sensorimotor, pre-operational, concrete operational, and formal operational and children aged 5 – 6 is in the second stage. Piaget believed that children in the first stage have ability to link numbers to objects (e.g., one dog, two cats, three pigs, four hippos). The child’s ability in this stage might be enhanced if he is allowed ample opportunity to act on the environment in unrestricted (but safe) ways in order to start building concepts [2]. This evidence suggests that children at the sensorimotor stage have some understanding of the Received date: 10/10/2016. Published date: 12/12/2016. Contact: Chu Cam Tho, e-mail: camtho@hnue.edu.vn 21 Chu Cam Tho, Nguyen Thi Quynh Anh concepts of numbers and counting. Educators of children in this stage of development should lay a solid mathematical foundation by providing activities that incorporate counting and thus enhance children’s conceptual development of number. The characteristics of pre-operational stage include an increase in language ability (with over-generalizations), symbolic thought, egocentric perspective, and limited logic. Thus, to teach numbers and numerical concepts effectively, teachers should use a tool as symbolization shows the relationship between them. Besides, teachers should contribute different learning styles in teaching such as visual, auditory, kinesthetic models to increase the effect of teaching and learning. 1.2. Introduction the number line as a tool teaching numerical contents One of the important and effective tools to show the order of numbers, the number line, could help teachers to teach numerical concepts as well as operations with natural numbers less than 20 for children early. In the paper of Heefer (2013), Heefer wrote “Despite its wide acceptance, the use of the number line in mathematics education is rather new [1]. It seems to have originated in the 1950s, Max Beberman, credited for many innovations in mathematics teaching, used the earlier term number scale”. As point out by Hans Freudenthal (1973), the number line should be used from the start of arithmetic at least, from very early on. In the beginning only natural numbers are noticed and marked on it; then with subtractions negative integers show up and are marked too; with dividing and shrinking the common fractions appear, and with measuring the decimal fractions, first only the finite, and then the infinite ones. Gradually the number line is filled, not with numbers and points, but rather with numerically – seized points [7]. Therefore, teaching number line for children aged 5 – 6 year old is totally conformable to the features of children’s cognitive development stages and help them to form and develop number sense. However, according to mathematics curriculum in Vietnamese, children learn about the number line when they start in grade 1 [13]. So, our group study named POMATH conducted a dependent experiment teaching some contents that used the number line as a tool for 5 to 6-year-old children (N = 152). The method which guides children to learn and use the number line as well as measure to assess the effectiveness of using the number line for the development of number sense of children aged 5 - 6 years old are presented immediately following. 2. Content 2.1. Measure using the number line to improve number sense for children aged 5 – 6 2.1.1. Propose main activities that will occur during each lesson Each lesson often took place in about one hour and included main activities: physical activities, playing math games and doing own book exercises. Table 1: Main activities that often occurred during a lesson Activities Purpose Time Suggestion Physical activities Helping children release energy and pays their attention on lesson better 5 – 10 minutes Keep one’s balance while standing on one foot; Dancing follow by sound of music or funny games in classroom, etc. 22 The proposal of using the number line to improve number sense for children... Playing math games Increasing interest in learning mathematics of children. 10 – 15 minutes Teachers prepared math games which are suitable to lesson such as game named “I need, I need!” when student learned numbers within 10. Do own exercise - book Helping children acquaint with formal learning mathematics activities before entry into primary school, such as hold a pencil, color a picture, read and write numbers, count,... 25 – 30 minutes Each child does their independent book exercises and teachers help them in need. 2.1.2. Design a list of exercises divided four level of competence The studies about education of UNICEF (The United Nations Children’s Fund) indicated that: “Every child is special, with unique combinations of abilities and needs that affect learning. And all children deserve the opportunity to learn in ways that make the most of their strengths and help them overcome their weaknesses” [17]. Following book named “A handbook for teachers of Multi-Grade Classes”: every child is an individual, with special social, emotional, intellectual, and physical qualities. Children are unique. They are individuals and no two children are alike: physically, emotionally, socially and intellectually, each child is a unique individual. Because children are unique, even if there are common needs and characteristics that children of a particular age or stage of development share, they must be understood by their parents and teachers in their uniqueness, and their individuality must be respected [10]. In addition, the recent results of study about mathematics education for children in Vietnam help us realize the important role of teaching and learning toward the common needs as well as consistent with individual ability of each child in the classroom [16]. It itself requires teachers, who immediate teach children, must prepare teaching plans carefully. However, with the features of classroom in a developing country as Vietnam: there are about 35 – 40, even 50 students in a class and material facilities are lacking, teachers have to face many troubles and it is difficult to take care of all children in class. Picture 1: The competence taxonomy To improve the effectiveness of teaching and help teachers overcome difficulties, we conducted the classification of students as groups based on their competency and building a list of numerical exercises accordant with each group, which also divided four levels following the Bloom’s taxonomy: A – knowledge, B – comprehension, C – low application, D – high application. To classify children before starting the course, they had to do a competence test. Then, the course syllabus for each child was conducted base on the result of their tests. The number line appeared in exercises as a tool to support the process of forming and developing number sense. 23 Chu Cam Tho, Nguyen Thi Quynh Anh 2.1.3. Building the communication between teacher-parent through the Contact Papers Picture 2: Example for Contact Papers It is evidently that increasing the communication teacher-parent and teacher-student is important contribution to improve the effect of teaching and learning. The studies of Matthew (2012) also indicated that “frequent teacher-family communication immediately increased student engagement as measured by homework completion rates, on-task behavior, and class participation” [12]. Thus, to maintain connection between teachers and parents, Contacts Papers were designed (see figure below). After each lesson, teacher had to write comments about learning contents, students’ behavior as well as marks they gained on the Contacts Papers and gave their parents. Parents were responsible for reading these comments and gave teachers feedback in order to create the best learning environment for children. 2.2. Experimental models of teaching number line for children aged 5 – 6 years old Two experimental teaching models were carried out at POMath centre: teaching in a group containing up to 5 students and teaching in class within one year, about one hour a week. 2.2.1. Purpose of experiment The purpose of the first teaching model is to assess the effect of putting the number line in the early preschoolers’ mathematics syllabus on children’s number sense development by case study method. Children allowed by their parents from different kindergartens were invited to join the experimental course. Children had to do a test before starting the experimental course. Then, we constructed course syllabus for each child based on the result of their test. The second teaching model is almost implemented in kindergartens to assess the practicability of the teaching model when put it on mathematics curriculum in kindergartens and we used common syllabus for all children in classes. 2.2.2. Contents of teaching experiment Picture 3: Example for teaching contents 24 The proposal of using the number line to improve number sense for children... Picture 4: Example for teaching contents We used number line in teaching the order of numbers and operations with numbers less than 20. Children were introduced to the number line as a symbol of numbers queuing from small to large through learning activities. It was to help children be more interested in our lesson and made the classroom atmosphere positive, which is very important for children. Firstly, they were acquainted with the order of numbers through activities such as standing in line, roll-call, joining up the numbers in increasing order to complete picture, etc. In the experimental process, we realized that the number line is very useful for teaching the relations of numbers such as number before or after a given number, numbers between two given numbers. For example: in order to find numbers between 4 and 8, firstly, teachers suggested children marked points to present 4 and 8 in the number line. And then, children found answers easily. Picture 5: Example for teaching contents Picture 6: Example for teaching contents Picture 7: Example for teaching contents Besides, the number line was also used to teach addition and subtraction numbers within 25 Chu Cam Tho, Nguyen Thi Quynh Anh 20. Teachers often created short stories about children’s problems in order to make their problem more interesting, more attractive. For example, “A lovely rabbit is standing at a point represented by 0 and he wants go to his school represented by 8 on the number line through 2 jumps. Look at this picture, do you know what rabbit jumped?”. Then, children started imagining and thinking about the picture and the story that had just been told. 2.2.3. Assessment After each lesson, teachers often recorded observations of children’s changes in the Study Diary. It helped both us and parents assess the progress of each child exactly and easily. Besides, children were also tested their number sense every 6 months. The questions were chosen carefully, focused on contents dealing with numbers, the relation of numbers, operations with numbers, etc. Teachers contributed by giving written test and oral tests in need. Example: Question 1: Sort given numbers in increasing value: 7; 3; 1; 5; 6 The purpose of this question is to assess children’s understanding about the order of numbers and sorting numbers skills. In order to answer this question, children can use an empty number line or use a sorting algorithm. Question 2: Rabbit’s house is represented by point 2, Squirrel’s house is represented by point 8 and their school is represented by point 6 on the number line. Whose house is closer to their school, rabbit’s or squirrel’s? Why? Picture 8: Illustration for question 2 Firstly, children needed to find the locations of rabbit’s house, squirrel’s house and their school on the number line. And then, children could estimate distance or count the number of unit segments between the house of animals and their school. Children could imagine that the number of unit segments is the number of footsteps that rabbits and squirrel have to go school on foot. 2.2.4. The result of teaching experiment In order to assess the effect of the experimental process, we conducted a survey about the feedback of children and their parents after the experimental course in two models. The result is approximately 100% children were interested in our lessons and 73% of their parents were quite proud of our syllabus and the learning result of their children. Besides, the following charts show the changes in the distribution of points before and after the experimental course. Children’s test points were classified as four levels: less than 4.5 points; from 4.5 points to less than 6.5 points; from 6.5 points to less than 8 points and more than or equal 8 points. It’s clear that the red and gold part in the chart describing the classification of test points after experiment is more than in the chart of before experiment. It indicates that there are more children whose points are more than or equal 6.5. Moreover, the number of student whose points are less than 26 The proposal of using the number line to improve number sense for children... 4.5 is reduced by 2%, from 4% to 2%. Specially, the results of some children showed significant progress, such as T.N, D.A, T.D, A.T, etc. Picture 9: The charts classify test points of before and after experimental course The following charts of before and after experimental course showed it clearly. Picture 10: T.N.’s result before and after experimental course Picture 11: D.A.’s result before and after experimental course 27 Chu Cam Tho, Nguyen Thi Quynh Anh 3. Conclusion This paper presents some reasons to teach number line for children aged 5 – 6 years old in Vietnam as well as describes our experimental teaching process. The experimental results show the effect of including the number line in preschoolers’ mathematics syllabus on the development of children’s number sense. However, children in the first model progressed much more than the second one. We also realized that experimental teaching process can’t be succeed if there is lack of enthusiastic and creative teachers and lack of good combination between teachers and parents. REFERENCES [1] Albrecht Heeffer, 2013. Algebraic symbolism as a conceptual barrier in learning mathematics. 6th East Asia Regional Conference on Mathematics Education, Phuket – Thailand. [2] Booby Ojose, 2008. 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