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
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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.
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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.
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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
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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
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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
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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. Applying Piaget’s Theory of Cognitive Development to Mathematics
Instruction. The Mathematics Educator, Vol. 18, No. 1, p.26–30.
[3] Carmel M. Diezmann, Tom Lowrie, Lindy A. Sugars, 2010. Primary students’ success on
the structured number line. Australian Primary Mathematics Classroom (APMC), 15 (4), pp.
24 – 28.
[4] Christopher T. Cross, Taniesha A. Woods, and Heidi Schweingruber, 2009. Mathematics
Learning in Early Childhood: Paths: Toward Excellence and Equity, Committee on Early
Childhood Mathematics; National Research Council.
[5] Department of education and early childhood development, 2009. Number line with whole
numbers.
[6] Diezmann, C. M. & Lowrie, T., 2006. Primary students’ knowledge of and errors on
number lines, In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds), Identities,
cultures, andlearning spaces. Proceedings of the 29th annual conference of the Mathematics
Education Research Group of Australasia, Vol. 1, pp. 171–178.
[7] Hans Freudenthal, 1973. Mathematics as an educational task. D. Reidel Publishing.
[8] Hien Dien Tran, 2013. Practising primary mathematics, Hanoi National University of
Education Publishing House.
[9] Janette Bobis, 2007. The empty number line: a useful tool or just another procedure?.
Teaching children mathematics.
[10] June Cezair Wallace, Hope MC Nish and John Allen, 1995. A handbook for teachers of
Multi-Grade Classes, Volume 2: Improving performance at the Primary level. UNICEF (The
United Nations Children’s Fund).
[11] Lago, Rachel M.; DiPerna, James Clyde, 2010. Number Sense in Kindergarten: A
Factor-Analytic Study of the Construct. School Psychology Review, Vol. 39 Issue 2,
p164-180. 17p.
[12] Matthew A. Kraft, Shaun M. Dougherty, 2012. The Effect of Teacher-Family Communication
on Student Engagement: Evidence from a Randomized Field Experiment. Journal of Research
on Educational Effectiveness.
[13] Ministry of education, 2010. Mathematics textbook grade 1. Vietnam education Publishing
House.
[14] Prentice Starkey, Alice Klein, 2008. Sociocultural influences on young children’s
mathematical knowledge. Information Age Publishing.
[15] UshaMenon, JodoGyan, 2012. Extending numbers with number sense. 12th International
Congress on Mathematical Education, Seoul – Korea.
[16]
[17]
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