CHAPTER 2
LITERATURE REVIEW
Introduction
This chapter presents the overview of research related to
the development of number sense studied by researchers, across the globe and
locally which served as guidelines for doing this research study grounded at the
primary school level. The different subtopics in this literature review include
the definition of number sense, the development of number sense among school
children, characteristics of number sense, behaviours that demonstrate number
sense and findings of some researches and studies done in the same area.
There were many studies
carried out on number sense in another parts of the world especially Australia,
Japan, United States, Taiwan and Sweden and few studies have been done in
Brunei by the officers from Curriculum Development Department, Ministry of
Education.
What is number sense?
The term “number sense”
is quite new in mathematics curriculum. Number sense is not easy to describe.
It is not a single entity, but rather involves a cluster of ideas such as
meaning of numbers, ways of representing numbers, relationships among numbers,
the relative size of numbers, and skill in working with them. Like “common
sense”, number sense is a valued but difficult notion to characterise
(McIntosh, A., Reys, B., Reys, R., Bana, J., and Farrell, B., 1997).
The NCTM document
Curriculum and Evaluation Standards for School Mathematics (2000) defines
number sense as the ability to decompose numbers naturally, use the
relationships among arithmetic operations to solve problems, understand the
base-ten number system, estimate, make sense of numbers, and recognise the
relative and absolute magnitude of numbers. Verschaffel, L. and De Corte, E.,
(1996) suggested that because of the complex, multifaceted and dispositional nature
of number sense it cannot be compartmentalized with special textbook chapters
of teaching units and that “the development of number sense results from the
whole range of activities of mathematics education, rather than a designated
subset of specially designed activities.” (McIntosh, et al 1997). This
definition of number sense suggests that teachers’ teaching should not merely
rely on textbook and workbooks. Rather teaching should go beyond that children
should be provided with the entire range of activities and kinds of instruction
that leads to a meaningful learning to develop number sense. Number sense
produces good and useful results with the least amount of effort (National
Research Council, 1989).
McIntosh, et al (1997) refers number sense as a person’s
general understanding of number and operations along with the ability and
inclination to make mathematical judgments and to develop useful and efficient
strategies for managing numerical situations. They pointed out that when
someone has number sense, it means that he or she possesses the ability to
understand, operate and understand the result of certain operations on numbers.
Development of number sense among school children
During the last 20 years, many important documents have
stressed the importance of developing number sense (Yang, D. C. & Reys, R.
E., 2001). It appeared that number sense has been brought in to the mathematics
curriculum many years ago, yet the presence has not been utilised by some
teachers. Despite its presence, administrators and teachers still put emphasis
more on memorized rules and algorithms. Instrumental instruction which results
in the rote learning of these traditional paper-and-pencil algorithms can
interfere with the development of number sense (Suydam, M. N. & Dessart, D.
J., 1976). Early introduction and practice of algorithms may legitimise a
single procedure and limit students’ computational fluency so that they cannot
choose methods that best fit the numbers or situations and hence interfere with
the development of number sense (Veloo, P. K. & Ali Hamdani, H. M. D.,
2004).
Markovits, Z. &
Sowder, J. T., (1994) observed that students who are taught in the
“traditional” way do not show understanding of number sense in problem solving
situations involving numbers. A number of mathematics educators seemed to agree
that the difficulties experienced by children in solving mathematics exercises
is closely related to the development of number sense thinking (Leutzinger, L.
P. & Bertheau, M., 1989; Burns,
M., 1989).
As number sense is not a single unit to be taught in
mathematics classroom, Yang, D. C. and Reys, R. E., (2001) pointed out that
number sense should be develops on a regular basis within each mathematics
lesson. The development of number sense results from a whole range of
activities that permeates the entire approach to the teaching of mathematics
(Greeno, J. G., 1991).
In developing number
sense as suggested by Tash, B., (in progress), teachers need to concentrate on
important mathematical concepts and shows in an exciting and fun “hands-on” way
of how to provide students a myriad of experiences necessary for them to
develop confidence with and understanding of numbers!!. These underpin teachers
to give students chance to experience games, manipulative and conceptual
activities that they love. This techniques and situations will be emphasized to
make mathematical concepts real and connected to the child’s world thus promote
the real learning of mathematics as Ekenstam, A., (1977) stated that without an
understanding of what numbers mean must present impossible barriers to learning
mathematics. Instruction that does not emphasize conceptual understanding but
rather emphasize algorithmic expertise can lead to the formation of defective
schema which once formed will interfere with the development of number sense
(Veloo, P. K. & Ali Hamdani, H. M. D., 2004).
Characteristics of number sense
It is not easy to define
the term “number sense” precisely, but broadly speaking, Bobis, J., (1996)
refers number sense as a well organised conceptual framework of number
information that enables a person to understand numbers and number
relationships and to solve mathematical problems that are not bound by
traditional algorithms.
The National Council of
Teachers (USA, 1989) identified five components that characterise number sense:
number meaning, number relationships, number magnitude and operations involving
numbers and referents for numbers and quantities. These skills are considered
important as they contribute to general feelings about numbers and lay the
foundation for more advanced skills.
A comprehensive
perspective of number sense, one proposed by the authors of Curriculum and
Evaluation Standards for School Mathematics (NCTM, 1989) pointed out that,
children with good number sense:
·
have
well understood number meanings;
·
have
developed multiple relationship among numbers;
·
recognize
the relative magnitude of numbers;
·
know
the relative effect of operating numbers;
·
develops
referent for measures of common objects and situation on their environment.
Shull (1998) offers
additional indicator of number sense; an intuitive understanding of numbers and
the effect of operations and numbers.
In a broader sense,
number sense is a well-organized conceptual network that enables a person to
relate number and its favourable operations.
Finding of some researches and studies on number sense
Studies conducted by
Veloo, P. K. & Ali Hamdani, H. M. D., 2004 in about 1200 secondary 1 and 2
students from 18 secondary schools throughout Brunei Darussalam found out that
boys performed better than girls in the number sense test. Similar studies done
by Noor Azlan, A. Z. & Munirah, G., (1993) on 406 ten year old children
from four different schools in Malaysia discovered that, in general, students seem
to experience difficulties with all the strands of number sense based on number
sense framework by McIntosh, A., Reys, B. J. and Reys, R. E. (1992) except
those related to the use of equivalent expressions and computing and counting
strategies. They noted that this may due to the fact that the other strands
require deeper understanding than just mechanical calculations.
The other study,
“Elementary school children’s number sense” by Menon, R., (2004) was with 750
elementary school children from Grades 4 through 7. The researcher found out
that as the number of years in schools increase, there is an increasing
reliance on algorithms and procedures. Overall, the elementary school children
from grades 4 through seven did not have a good grasp of number sense.
Results from previous
studies shows that majority of school children do not have number sense as
shown from the low scores in administered tests. The level of number sense
among the sample of students investigated was low. The findings will inform
administrators, mathematic teachers and curriculum developers to better
appreciate how students’ learn and think when dealing with mathematics
problems. Further research is needed to address this problem.
Summary
Through the supports, suggestions and recommendations
reported in this chapter encouraged the writer to develop a research study on
finding the present level of number sense among some of the primary school
pupils in Brunei and to find out whether there would be any significance
difference in performance by gender.
Since the term “number
sense” is relatively new to the mathematic curriculum in Brunei, therefore not
many studies have been done. There was one study which relates to the writer’s
study but only involved in secondary schools in Brunei. Hopefully, this study
which involves primary school pupils could answer questions regarding the
current level of pupils’ number sense and it also hope that the findings of the
study will encourage mathematic teachers to change classroom norms and
traditions and to integrate activities that lead to the development of number
sense.