CHILDRENS' NUMBER SENSE
Ahmad Zanzali, Munirah Ghazali
Faculty, School of Educational Studies, Universiti Teknologi
Malaysia. Universiti Sains Malaysia.
reported here is part of a bigger study that aims to look at
children’s’ notion about number sense. This study attempt to assess
chil dren’s number sense related to five strands of number sense
based number sense framework by McIntosh, Reys and Reys (1992). The
five strands are: understanding the meaning of size and numbers,
understanding and use of equivalent forms and representation of
numbers, understanding the meaning and effect of operations,
understanding the use of equivalent expressions and computing and
We are also interested to find out whether
children demonstrate understanding of numerical situations in which
they solve computational problems; whether there is a relationship
between the strategies children use to solve number problems and
number sense. Two tests were administered on 406 ten year old
children from four different schools in Malaysia. The first test
attempted e test to assess students’ understanding of number sense
with respect to the five strands of number sense. The second test is
a written computation test with similar items designed in number
sense format. A series of interviews were also conducted to discover
the relationship between understanding of number concepts and the
The results obtained indicated there is a
wide range of performance. It is discovered that, in general,
students seem to experience difficulties with all the strands of
number sense except those related to the use of equivalent
expressions and computing and counting strategies. This may due to
the fact that the other strands require deeper understanding than
just mechanical calculations. The result from the written
computation test show that generally, students perform better on
written computation test than on similar items in number sense
primary school mathematics curriculum clearly states that the main
of the curriculum is to build and develop children’s understanding
in the number concept and at the same time attains high facility in
the basic skills. Both these are to be applied to problem solving
situations, mostly related to everyday problems (Noor Azlan, 1993).
Through these, children are expected to appreciate the beauty and
importance of mathematics. Mathematics is seen as a branch of
knowledge that is useful in dealing with everyday life problems in a
disciplined manner parallel with demands of society in a developed
nation. (Kementerian Pendidikan, 1989). Hence, the teaching and
learning of mathematics, even at the early stages, should attempt at
instilling critical and creative thinking, apart from memorizing
mathematical algorithms and procedures (Tajul Arrifin and Nor’ Aini,
concepts in primary school mathematics
curricular reform documents (such as National Council of Teacher of
Mathematics, 1989;Australian Education Council, 1991;Cockcroft,1982,
Kementerian Pendidikan, 1991) emphasize the importance of number
sense based on the rational that numbers sense will be very helpful
to understand numbers in general. Relatively, the focus on term
“number” in the mathematics curriculum is quite recent and most has
targeted their arguments to the primary school level (Burns,1994;
Hiebert,1984; Plunkett,1979; Skemp,1982, Kementerian Pendidikan,
1989). The understanding of numbers is of fundamental importance in
primary school mathematics. The understanding and attainment
of these basic skills must be constantly applied to real life
problems (Noor Azlan, 1993).
Attaining number sense becomes more
important when they proceed to secondary school as the secondary
school curriculum are based on three strands, that is number,
shapes and relation(Kementerian Pendidikan, 1989).The question
then, is have the children mastered number sense well enough so as
to be able to grasp the content of the secondary school syllabus? In
general, only a limited number of students do really understand
number sense while solving problems (Markovits & 31 Sowder,1994;
Noor Azlan and Lui; in preparation). Further, Markovits & Sowder
(1994) observed that students who are taught in the “traditional”
way do not show understanding of number sense in problem solving
situations involving numbers (see also Reys,1973). 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 &
A survey of
literature indicated that number sense is difficult to define and
that it is not a single entity, but rather has many dimensions. Like
‘common sense”, number sense is a valued but difficult notion to
characterise (McIntosh, Reys, Reys, Bana and Farrell, 1997).
Verschaffel and De Corte (1996) emphasize that “this complex,
multifaceted, and dispositional nature of number sense suggests that
it cannot be compartmentalized into special textbook chapters or
instructional 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, Reys, Reys, Bana and Farrell, 1997). Nonetheless, various
‘indicators” of number sense have been hypothesized (McIntosh, Reys,
Reys, Bana and Farrell, 1997). These include well understood number
meanings, existence of and reliance on multiple numerical
relationships, recognition of relative magnitude of numbers,
awareness of the relative effect of operating on numbers, and use of
referents for measures of common objects and situations and in their
environments (NCTM, 1989). Shull (1998) offers additional indicator
of number sense; an intuitive understanding of numbers and the
effect of operations and numbers. It is a well organized conceptual
network that enables a person to relate number and operation
propitious. Number sense is characterized by an individual’s ability
to use his/her understanding of mathematics in flexible and creative
ways to make mathematical decisions and to develop useful strategies
for handling numbers and operations. In short, number sense refers
to the ability to use numbers and quantitative methods as a means of
communicating, processing and interpreting information (McIntosh,
Reys, Reys, Bana and Farrell, 1997). It refers to the understanding
about numbers and their related mathematical operations and the
ability (tendency) to use this understanding to make decisions about
mathematically related situations.
can be seen as carefully arranged concepts that allows one to relate
between properties of numbers with that of operations (Sowder,
1992). It can be identified as the ability to synthesize numbers and
at the same time able to recognize its representations. Number sense
also involves the ability to compare numbers, to sequence numbers in
meaningful forms, relate the values the numbers
compute mentally, and be able to sue the appropriate strategy to
understand the impact of certain operations. Number sense refers to
the ability to understand, operate and understand the result of
certain operations on numbers.
Number sense and written computation
argues that children must understand number meanings if they are to
make sense of the way numbers are used in real life situations.
Mathematics educators are concerned that many students demonstrate
little understanding of numerical situations in which they solve
number problems. The lack of number sense seems to be the result of
mindless application of the standard written algorithms which they
learned in school (Yang, 1997). Students are good rule followers,
however, they do not always understand they procedures they learned
(Hiebert, 1984). Students are often better at manipulating and
following symbol rule than they are at making sense of the numerical
situations. Despite efforts by the mathematics education community
to move away from the traditional conception about mathematics,
recent findings (Noor Azlan and Lui, in preparation), have strongly
indicated that most children have not attained the understanding
that demanded by the new curriculum (Noor Azlan Ahmad Zanzali, 1995;
Noor Azlan and Lui, in preparation).
and Reys (1992) developed a framework for examining number sense
based on studying and reflecting on the literature associated with
number sense, estimation and mental computation. The framework
formulated six number sense strands:
understanding and use of the meaning and size of numbers
understanding and use of equivalent forms and representations of
understanding the meaning and effect of operations
understanding and use of equivalent expressions
and counting strategies.
of the study
Based on the
above discussion, there is a need then to:
children's’ number sense to the first five strands in the number
strand number five in the number sense framework. The sixth
strand, though is as important if left for further or subsequent
whether children demonstrate understanding of numerical
situations in which they solve number problems, whether there is
a relationship between the strategies children use to solve
number problems and number sense.
the qualitative relationships between number sense and the
ability to compute numerical problems.
The sample of
study consisted of children in year 4 of the primary school. Two
tests were administered. This is further followed by interviews on
All test items
and directions were presented in the national language. A 47 item
test (adapted from McIntosh et. al.; 1997) containing various
aspects of number sense were given to the children to solve. This is
followed by a 15 written computation test. All the 15 written
computation items are constructed in number sense format and
included in the number sense test. For example, question 36 in the
number sense test is:
calculating an exact answer, circle the best estimate for 5/6 + 8/9
A. 1 B.2 C.19
Question 1 on
the written computation test is:
which is the number sense item could be solved by thinking that each
of the two fractions are slightly less than one, therefore the sum
of the two fractions would be almost like the sum of 1 and 1 that is
choice B. It is hoped that the children can assess the suitability
of their answer based on their understanding of fraction size and
the operation involved. Question 1 on the computation test requires
student to actually do the calculation in order to find the answer.
Table 1 below gives the summary for items in number sense test and
items in written computation test.
38 Table 1.
Summary for items in number sense test and items in written
Number Number sense strand number of
Nb. Bold items
have similar items in computation format.
S9 - S17
S18 – 21,
K5, k2, k11,
K3, k6, k7,
k9, k10, k12, K4
of the findings
The items in
both the number sense and written computation test were given 1 for
a correct score and 0 for an incomplete or incorrect answers. The
percentage of correct answer for the number sense test is 36.9%. The
percentages of correct answers for the various strands of number
sense differ as indicated in the table below.
Percentages of correct answers for the five strands of number sense
and the range of marks for each strand.
Strand Percentage of Correct answers
Range of mark
Representation 31.7 100
Operations 29.2 88.9
Expressions 45.5 91
Computing Strategies 41.2 100
The data from
this study shows that students perform better on number sense
strands 4 (equivalent expression) and on strand 5 (counting and
computing strategies). Students seem to have difficulty with the
strands number concepts, effect of operations and multiple
representations. This may be due to the fact that these three
strands require deeper understanding than just mechanical
calculations. Despite the relatively “low” percentage of correct
answer for each strands of the number sense, the range of mark for
each strand is more than 90%. This shows that they are students who
have well developed number sense in each of the strand of number
sense tested as well and at the same time we as students who have
very poor understanding of each strand of number sense.
this indicates that the children in this study have the most
difficulty with the number sense strand “effect of operations”.
Understanding the meaning and effect of an operations, either
generally or as related to a certain set of numbers (eg. division
means breaking a number into a specified number of equivalent
subgroups, or multiplying by a number less than 1 produces a product
less than the other factor) includes judging the reasonableness of a
result based on understanding the numbers and operations employed.
There were nine questions in the number sense test on children’s’
understanding of effect of operation. The questions focussed on the
effect of operation instead of asking children to actually carry out
the computation. For example;
Q. 30) Which
of the following will give the highest value?
29 ¸ 0.8 B. 29 x 0.8 C. 29 + 0.8 D. Impossible
requires the children to choose an operation that will produce the
highest value as the answer. The percentage of correct answer for
this question is only 4.4%. The researchers wanted to know whether
the children in this study have the ability to understand the effect
of different operations on the same operand. In this study, 63.5
percent of the children chose B or 29 x 0.8 as an operation 38 that
will produce the highest value while only 4.4 percent of the
children chose A (29¸ 0.8) which is the correct answer. 10.6
percent of the children chose C while 18.5 percent of the children
said that it is impossible to get the answer without actually
carrying out the calculation. The choice is based on the assumption
that the product of multiplication is greater and that the final
result of a division will produce a smaller value. One implication
from this study is that in order for mathematics to be meaningful to
children, they have to understand the effect of numerous operations
and calculations that they carry out besides just striving to get
the correct answer for the calculations.
of correct answer for computation test is 61.4%. The percentage of
correct answer for computation test is better than the percentage of
correct answers for number sense test. Table 3 below gives a
comparison of student’s performance on items in number sense format
and the same item in computation format.
Comparison of student’s performance on items in number sense format
and the same item in computation format.
Percentage of correct answer
Number sense test
test – number sense test)
S36 K1 17.9
S46 K15 29.9
S45 K14 30.3
S47 K10 53.5
S34 K3 74.4
S38 K6 6 85.9
S22 K4 32.1
S40 K9 37.1
S41 K10 17.4
S43 K12 81.2
S44 K13 72.4
S42 K11 47.7
S26 K5 17.7 0
S37 K2 37.9
S39 K7 12.4 6
From table 3,
it can be seen that children have difficulty solving problems in
number sense format. For example the percentage of correct answer
for question 36 which is written in number sense format is 17.9%.
The percentage of correct answer goes up to 70.9% when the same
question is written in computation format. The percentage of correct
answers in computation format is higher in 11 out of the 15
questions given. Question 36 in computation format (5/6 + 8/9) is
considered a complex addition of fraction for 10 year old children.
However, more than 70% of the children correctly computed the exact
answer for this item. When the question is presented in the number
sense format, only 18% of the children could get the correct answer.
The majority of the children (67.9%) thought that the answer to 5/6
+ 8/9 is either 19 or 21. This suggests that most of the children
were unable to make sense of the relationship between the fraction
size and the operation involved. Even though most students can
correctly calculate the answer, it is still questionable whether the
children demonstrate understanding of the numerical situation.
This study has
generated some interesting findings worthy of further probe. We are
now in the midst of conducting interview sessions with selected
students. The interviews will center on the relationships between
the ability to compute numerical problems and number sense. Why does
there exist a gap between ability to calculate using established
algorithms and attaining number sense. In general children are able
to do very well when asked to calculate but do not seemed to show
understanding on the concept of numbers. Current changes in the
mathematics curriculum require children to attain higher order
thinking. The learning of mathematics should be more focussed on
thinking and reasoning about mathematics rather
memorizing the algorithms mechanically. Data from this study shows
that children have difficulty understanding basic number concepts as
shown with the low scores in tests of number concepts particularly
those related to the effects of operations and multiple
representations. The study also shows that children perform better
on items that are written in computation format as compared same
item written in number sense format.
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