Assessing Students’ Number Sense: What to Be Considered?
Susilahudin Putrawangsa
1
, Erpin Evendi
1
and Uswatun Hasanah
2
1
Mathematics Education Department, Universitas Islam Negeri Mataram, Jalan Pendidikan, Mataram, Indonesia
2
Department of Informatic, STMIK Bumigora, Jalan Ismail Marzuki, Mataram, Indonesia
Keywords: Operations, Mathematical Judgment, Unitizing, Magnitude, Cardinality, Number System, Estimation,
Subitizing, Screening, Measurement, Assessment.
Abstract: Since the notion of number sense is remain unclear among researchers, the recent study intends to provide a
literature-based understanding of the key concepts of number sense and how to assess the notion. Although
researchers define the term ‘number sense’ in many different ways, they somehow agree that the term
generally depicts someone’s proficiency in dealing with numbers and number operations in computational
situation or problems. People who have an adequate sense of numbers possess: (1) an appropriate
understanding of numbers and how they are related one to another; (2) an understanding the meaning of
operations and how they are related and impact numbers or other operations; (3) a capability performing
computation by utilizing their understanding of the numbers and the operations fluently and flexibly in dealing
with number-related problems including making reasonable estimation; and (4) a faculty in making
appropriate judgment of calculation including identifying potential error of a computation and making
estimation. Therefore, assessing students’ number sense means assessing their faculties on the four aspects.
1 INTRODUCTION
The term number sense is used to depict a complex
idea referring to the extent of people aptitude in
working with numbers and their manipulation
(operations). Due to its complexity, there is no any
single definition yet that are used among scientists to
represent the term ultimately. It is considered a
holistic construct that is not easy to define in a single
definition (Jordan & Dyson, 2016; Yang & Wu,
2010).
However, scientists strongly agree that the faculty
in dealing with numbers will determine people
aptitude in the higher aspects of mathematics and also
determine their future career (Jordan & Dyson, 2016;
Wu, 1999; Jordan et al., 2013; Vukovic et al., 2014;
Ontario, 2013; Cai & Knuth, 2011; Bill, et.al. 2010;
Hope & Sherrill, 1987; Bobis, 1991; Case & Sowder,
1990; Cobb, et.al., 1991; Jordan et al., 2009). It is
claimed that students’ lack competency in dealing
with numbers affects their achievements in the other
aspects of mathematics, such as working with
fractions and algebra (Wu, 1999; Jordan et al.,
2013;Vukovic et al., 2014), dealing with algebraic
problems (Ontario, 2013; Cai & Knuth, 2011; Bill,
et.al. 2010), doing mental calculation and estimation
(Hope & Sherrill, 1987; Bobis, 1991; Case & Sowder,
1990), and performing problem solving (Cobb, et.al.,
1991; Jordan et al., 2009). In another aspect, students’
level of number sense can be used as the basis to
predict students’ achievement in higher aspects of
mathematics (Robinson, Menchetti, & Torgesen,
2002).
Considering the potential impacts, assessing
students’ number sense is crucial to prevent them
from the failure in mathematics. In this case, Jordan
& Dyson (2016) stress that screening students’
aptitude towards numbers can be used as the basis to
determine students who need more assistance to
prevent them from failure in learning mathematics.
However, what and how to assess students’
number sense is still debatable among scientists due
to its broad and complex ideas (Jordan & Dyson,
2016). Therefore, the current article intends to discuss
the key concepts of number sense and how to assess
the notion.
2 METHOD
Since the term number sense refers to complex ideas
relating to human capability in dealing with numbers
188
Putrawangsa, S., Evendi, E. and Hasanah, U.
Assessing Students’ Number Sense: What to be considered?.
DOI: 10.5220/0008519201880197
In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 188-197
ISBN: 978-989-758-407-7
Copyright
c
 2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

and number manipulations, there is no single definite
definition among scientists to describe the term. For
example, among psychologists, number sense is
defined as the ability to recognize, understand,
estimate and work with numbers regarding non-
symbolic ideas of numbers (see Dehaene, 2011).
Meanwhile, educators tend to regard number sense as
the faculty in dealing with not only non-symbolic
aspects of numbers but also the symbolic one in
number-related problem solving (see NCTM, 2000 or
Fosnot & Dolk, 2001). The slightly different views
among the two domains of scientists may contribute
to confusion among researchers and educational
practitioners, especially those beginners, in
conceiving and assessing students’ number sense.
Regarding the proposition above, the current
article intends to build a robust literature-based
explanation about the definition and considerations in
assessing students’ number sense. There are three
main intentions of the study, such as (1) constructing
literature-based understanding relating to the views
among scientists about number sense; (2) clarifying
elements in assessing students’ number sense; and (3)
identifying considerations in assessing students’
number sense.
There are seven sources of qualified literature
which are studied intensively to identify their
definitions, key concepts, and characteristics of
number sense to uncover potential similarities and
differences. The similarities and differences then are
studied critically in order to identify their common
tendencies. The literatures are from various type of
publications (such as journal articles, books, book
chapters, and a proceeding) which is publisher by
qualified publishers (including Elsevier Academic
Press, Oxford University Press, other institutional
publishers) from various countries (like The United
State of America, The United Kingdom, Sweden, and
Indonesia) and recent years of publications (range
from 2000 to 2018).
Before concluding, the initial findings are
presented and discussed among experts and educators
to be criticized to gain a robust conception about the
object of the study. The conclusion is discussed in the
recent article.
3 FINDING AND DISCUSSION
In this part, various definitions of number sense, its
key concepts and its indicators of assessments are
discussed which based on several related and
qualified literature from several different resources
and recent years of publications.
3.1 Definitions of Number Sense
No doubt among the scientists that students’ sense of
numbers will determine their aptitude in the higher
aspects of mathematics and also determine their
future career (Jordan & Dyson, 2016; Wu, 1999;
Jordan et al., 2013; Vukovic et al., 2014; Ontario,
2013; Cai & Knuth, 2011; Bill, et.al. 2010; Hope &
Sherrill, 1987; Bobis, 1991; Case & Sowder, 1990;
Cobb, et.al., 1991; Jordan et al., 2009). Regarding the
significant impact of number sense, it is necessary to
have a clear and appropriate understanding of the
concepts underpinning the terms.
The term ‘number sense’ is a complex concept
that does not easy to define precisely. In general, the
extent of students’ flexibility in dealing with
quantitative ideas about numbers, their relationships
and as well as their manipulations indicates students’
sense of numbers. There are various definitions used
to express such a kind of sense. Jordan, Fuchs &
Dyson (2014), for example, merely define number
sense as children knowledge of numbers, number
relations, and number operations. Meanwhile,
Andrews & Sayers (2014) relate number sense as the
ability to operate flexibly with number and quantity.
Some other researchers assert number sense in the
broader context that seeing number sense is not only
a form of knowledge (which to possess or not) but
also a type of problem-solving expertise that develops
over time as the result of experiences. Fosnot & Dolk
(2001), for example, consider number sense as
proficiency in grasping the notions of numbers and
their manipulations in emerging and selecting
efficient counting strategies in problem-solving
contexts. Fosnot & Dolk (2001) stress that students’
senses of numbers are developed over time through
arbitrary or nonlinear learning trajectories (as
opposed to linear learning trajectories that are
traditionally practiced in mathematics instructions) as
the result of intended or hypothesized learning
experience. Here, students’ development of senses of
number forms a landscape of learning containing
some big ideas (such as cardinality and magnitude of
numbers), strategies of reasoning (such as one-to-one
tagging, skip counting, and counting on), and models
of thinking (such as tallies and number lines). The
landscape of learning is not developed instantly but it
is developed over time through hypothesized learning
trajectories (see Simon’s idea of hypothetical learning
trajectories (Simon, 1995)).
Moreover, Fosnot & Dolk (2001) assert that
calculating with number sense means the ability to
select appropriate and efficient counting strategies by
considering the characteristic of the numbers and the
Assessing Students’ Number Sense: What to be considered?
189

problems. To be able to perform in this level, students
are required to have a well-established understanding
of numbers, operations, and the relationship between
and among them. Then, those understanding are
utilized in emerging efficient counting strategies
flexibly in solving number-related problems. To
easily understand the idea, Fosnot & Dolk (2001)
confront between counting by numbers sense and
non-number sense as they said “Using algorithms, the
same series of steps with all problems, is antithetical
to calculating with number sense. Calculating with
number sense means that one must look at the
numbers first and then decide on a strategy that is
fitting—and efficient.” (Page 124).It means that one
a student has appropriated or adequate number sense,
he/she does not rely on a specific counting procedure,
but he/she has flexibility in emerging and selecting a
counting strategy that seems more efficient to tackle
computation problems.
Some scientists elucidate number sense in a more
operational definition. NCTM (2000), for example,
describe students who proficient in numbers as they
can decompose numbers naturally, use particular
numbers as referents, use the relationships among
arithmetic operations to solve problems, understand
the base-ten number system, estimate, make sense of
numbers, and recognize the relative and absolute
magnitude of numbers. NCTM (2000) proposes that
teaching for numbers and operations should focus on
building students’ number sense consisting of three
competencies. The first is the understanding of
meaning the numbers and the relationships among
them. The second is the comprehending with number
operations and how they are related one to another.
The third is handy in computation and estimating by
employing their understanding of numbers and
operations. Therefore, broadly speaking, NCTM
(2000) defines students’ number sense as students’
ability in utilizing their understanding of numbers and
operations in dealing with computational problems.
Other researchers relate students’ number sense
as the ability to think logically, critically and
creatively in dealing with computational problems.
Putrawangsa & Hasanah (2018), for example,
consider number sense as the ability to utilize logical,
critical and creative reasoning and understanding
about numbers and operations in dealing with
number-related problems flexibly, effectively,
efficiently and practically. Once students possess an
adequate sense of number, they can flexibly
formulate by themselves several counting strategies
and logically and critically select one of those that fit
and efficient with the problems. In line with Fosnot &
Dolk (2001), Putrawangsa & Hasanah (2018) stress
that students who possess a sense of number do not
rely on a specific counting procedure or an algorithm,
but they can emerge various counting strategies or
counting approaches to handling a particular counting
problem.
Dissimilar from other researchers, Ali (2014)
incorporates the notion of number sense with the
ability to produce mathematical judgment about
numbers and their operations. Ali (2014) stresses that
students with good number sense can use numbers in
flexible ways to make a mathematics-based judgment
and to develop useful strategies for handling
problems relating to numbers and operations. Here,
students who have a sense of numbers can compose
their procedures in doing the calculation and
recognize significant numerical errors of a particular
calculation. They own a good sense of numerical
magnitude which helps them in represent the same
number in multiple ways. Ali (2014), moreover,
asserts that students with good number sense have the
ability to use numbers in flexible ways to make
mathematical judgment and to develop useful
strategies for handling numbers and operations
Considering the various depictions of number
sense among scientists describe above, all researchers
relate number sense to the idea about numbers and
their manipulations (operations). However, there are
some varieties among them in expressing the focus
and the scope of the idea. Some of them relate it to
the context of problem-solving, making mathematical
judgment, or dealing with computation. Hence, it can
be deduced that the notion of number sense is used to
describes the extent of students’ proficiency in
dealing with quantitative ideas of numbers, their
relationships, and their manipulation (operations) in
context of solving number-related problems
including making estimation and number-related
judgment. Students who have a strong sense of
numbers can be characterized as the following
evidences: (1) own understanding about numbers,
their manipulations (operations), and the
relationships among them;(2) unbounded to a certain
counting procedure or algorithm; (3) being able to
generate various counting strategies by their own
flexibly in dealing with number-related problems; (4)
being able to make number-based estimation and
judgment. The skill develops over the time as the
result of the development of students’ understanding
of numbers, their manipulations (operations), and the
relationships among them acquired from life
experiences, both educational and non-educational
experiences.
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3.2 Various Perspectives toward the
Domain of Number Sense
It is identified that there are two main perspectives
regarding numbers which shape the scope of the
discussion about number sense, such as non-symbolic
and symbolic aspects of numbers (Sokolowski &
Ansari, 2016). The non-symbolic aspects refer to
human aptitude about quantitative ideas of numbers
without considering written or verbal representations
of numbers (symbolism). For instance, children who
are still illiterate can recognize one box which
contains more marbles. This proficiency is human
inherent capacity, and it is developed naturally.
Meanwhile, symbolic aspects of numbers refer to
human faculty in working with quantitative ideas of
numbers that implicate numerical representations
either in the form of written or verbal expression. For
example, children know that five is used to represent
five things or understand that the expression 5 – 2
means five is subtracted by two. People who grasp
advanced concepts about symbolic ideas of numbers
can easily recognize the meaning of complicated
numerical symbolization. Brain representations that
underlie non-symbolic numbers are considered innate
in the human brain and consequently are not
necessary to be constructed through learning
interventions (Cantlon, 2012). In contrast, brain
representations that underlie symbolic numbers are
not innate, and therefore they must be developed
through learning interventions (Ansari, 2008).
Regarding this symbolization, number sense has
defined the difference between psychologists and
educators slightly. Cognitive psychologists tend to
consider number sense as the ability to recognize,
understand, estimate and work with non-symbolic
ideas of numbers (Dehaene, 2011). Meanwhile,
educators tend to regard number sense as the capacity
to deal with both non-symbolic and symbolic ideas of
numbers (NCTM, 2000; Fosnot & Dolk, 2001).
NCTM (2000), for example, stresses that
understanding numbers and their symbolizations are
the basis for developing students’ number sense. The
similar findings also coined by Berch (2005) that
psychologists and mathematics educators work to
different definitions of number sense. However, both
groups of researchers agree that developing students’
number sense in early grades is crucial since it plays
important role as foundational concepts for students
to master further advance concepts of mathematics
(Jordan et al. 2013) and potentially determine
students’ opportunity in the future career (National
Mathematics Advisory Panel, 2008).
Another perspective relating to number sense is
the one proposed by Jordan & Dyson (2016). They
use the term verbal and non-verbal to refer two
aspects of number sense. The term verbal knowledge
of numbers refers to students understanding of
symbolic aspects of numbers. Meanwhile, for the
non-symbolic aspects, they coin the term non-verbal
knowledge of numbers. They argue that both verbal
and non-verbal knowledge of numbers is prominent
to support students’ sense acquisition of numbers.
If the previous perspectives of numbers sense
differentiate number sense regarding number
representations. Andrews and Sayers (2014) propose
a distinct perspective in looking at number sense
which is based on its utility. Andrews and Sayers
(2014) propose the term foundational and applied
number sense. The foundational number sense is
defined as students’ early number-related
understanding as the result of early formal
instructions. Meanwhile, the term applied number
sense indicates students’ proficiency in utilizing their
foundational number sense to function effectively in
society. According to Andrews and Sayers (2014), the
foundational sense of numbers at least comprise
seven aspects, such as (1) involving number
cognition, its vocabulary and its meaning; (2)
incorporating systemic counting, including the notion
of cordiality and cardinality; (3) Recognizing the
relationships between numbers and quantity; (4)
Recognizing the magnitudes of numbers and the
comparisons between those magnitudes; (5)
Recognizing and making estimation of numbers; (6)
Being able to perform simple arithmetic operations,
such as addition and subtraction; and (7) Recognizing
of number patterns. Other than those seven aspects
may be considered as applied number sense. Students
who own an excellent applied number sense will
consider the relationship among operated numbers
and the operations in inventing and selecting counting
strategies, consider the context of the problem in
inventing the strategies, recognize the unreasonable
answer, and recognize the need for making the
estimation. However, the differences between
foundational and applied number sense remain vague.
Researchers’ and educators’ view toward number
sense definition and perspective will determine their
approach of assessing students’ number sense.
3.3 Indicators of Number Sense
Assessment
Many researches identify that students’ number sense
will determine their mastery in the higher aspects of
mathematics and also determine their future career
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191

(Jordan & Dyson, 2016; Wu, 1999; Jordan et al.,
2013; Vukovic et al., 2014; Ontario, 2013; Cai &
Knuth, 2011; Bill, et.al. 2010; Hope & Sherrill, 1987;
Bobis, 1991; Case & Sowder, 1990; Cobb, et.al.,
1991; Jordan et al., 2009). Therefore, assessing
students’ number sense is crucial to prevent them
from the failures. However, what are the
considerations once assessing students’ number
sense? This question will be addressed in this part.
According to NCTM (2000), the central core of
teaching numbers and operations is teaching for
number sense. Once students acquire an adequate
sense of numbers, they can decompose numbers, use
particular numbers as referents, use the relationships
among arithmetic operations to solve problems,
understand the base-ten number system, make an
estimation, and recognize the relative and absolute
magnitude of numbers.
NCTM (2000) formulates several main
competencies about numbers and operations that
comprise students’ number sense. Those are
understanding the meaning of numbers, ways of
representing numbers, recognizing the relationships
among numbers, number systems, understanding the
meanings of operations, how the operations are relate
done to another; being able to compute fluently and
making reasonable estimation in computation. Those
competencies can be categorized into three aspects.
The first aspect is dealing with understanding
numbers including the relations among the numbers.
The second is the understanding of operations and
how they are related one to another. The last aspect is
relating to skillfulness in performing computations
fluently and flexibly by utilizing their understanding
of numbers and operation in number-related problem-
solving context. Making estimation is also a part of
this skill. In overall, according to NCTM (2000),
students’ number sense can be determined by
investigating students’ (1) understanding of numbers
(meaning of numbers); (2) understanding of ways to
represent numbers; (3) understanding of the
relationship among numbers; (4) understanding of
number system; (5) understanding of the meaning of
numbers operations; (6) understanding of the
relationship among operations; (7) ability in compute
fluently; and (8) ability in making reasonable
estimation (see Table 1).
A slightly different perspective proposed by
Fosnot & Dolk (20010. They argue that once students
acquire senses of numbers, they could decide
computational strategies that fit with a problem being
dealt with among other strategies in their repository
(Fosnot & Dolk, 2001). In deciding the strategies,
they consider the characteristics of the problems (the
context), the involved numbers and the operations,
the relationships within the numbers and the
operations, and the relationships between numbers
and operations. Here, students’ number sense is built
upon a complex system of cognitive development
about numbers, operations, and how they are related.
That comprehensive understanding leads to the
development of big ideas, strategies, and models of
thinking toward numbers and operations which help
them to deal with computational problems fluently
and flexibly. Moreover, Fosnot & Dolk (2001) assert
that students who have a well-developed number
sense do not rely on a specific static rote algorithm or
procedure in dealing with computational problems.
Instead, they rely on their mental understanding about
the problems (mental mathematics) to emerge various
counting strategies and then intelligently decide the
strategy among the emergent strategies that are
considered to be the most efficient and fit with the
problems. They even can provide multiple
perspectives or approaches of solutions for a single
computational problem.
Fosnot & Dolk (2001) claims that some big ideas
that are crucial in the early development of students’
numbers sense. In the early development students’
number sense, students should be supported to
develop (1) their understanding of numbers
(including the idea of one-to-one correspondence
when counting, cardinality, magnitude, hierarchical
inclusion, compensation, and part/whole
relationship), (2) their understanding of number
system (including the idea of unitizing, ten-based
unitizing, and place value), (3) their understanding of
basic facts of addition and subtraction, such as ten-
based combination, commutative property, doubling ,
swapping, constant difference, and cancelling. Those
big ideas of early number sense can be utilized as the
basis to clarify the extent of students’ sense of
numbers.
Another researcher, such as Andrews & Sayers
(2014), formulates elements building students’
foundational number sense. The elements comprise
the following notions, such as (1) number cognition,
its vocabulary and its meaning; (2) systemic counting,
including the notion of ordinality and cardinality; (3)
the relationships between numbers and quantity (a
number represents a certain quantity); (4) the
magnitudes of numbers and the comparisons between
those magnitudes; (5) making estimation of numbers;
(6) performing simple arithmetic operations, such as
addition and subtraction; and (7) number patterns.
Those notions can be considered as indicators once
assessing students’ early number sense.
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Meanwhile, according to Ali (2014), students with
strong number sense are indicated by proficiency in
(1) using numbers in flexible ways when adding,
subtracting, multiplying or dividing; (2) using
benchmarks to make mathematical judgments; (3)
making mental calculations and reasonable
estimations; (4) making predictions; (5)
understanding numerical relationships between
mathematical concepts, facts and skills; and (6)
recognizing unreasonable answers.
Both Jordan, Fuchs, & Dyson (2014) and Jordan
& Dyson (2016) state that students’ early number
sense can be indicated through their understanding on
the three core areas, such as the notions of numbers,
number relations, and number operations. The
notions of numbers include the concept of
subitization, cardinality, and the symbolism of
numbers. Meanwhile, the notion of number relations
comprises students’ understanding of the magnitude
of numbers and conceptual structure of numbers
(mental structure of numbers). Understanding basic
facts of addition and subtraction and the ability to
decompose numbers in various forms are two main
notions in the early development of number sense
relating to early number operation proficiency.
Meantime, Putrawangsa & Hasanah (2018) regard
number sense as the ability to reason with numbers.
They define number sense as the ability to utilize
logical, critical and creative reasoning and
understanding of numbers and operations in dealing
with problems relating to numbers flexibly,
effectively, efficiently, and practically. A student who
has a good number sense has adequate understanding
and skillful in the aspect’s numbers, number
operations and calculations. In the aspect of numbers,
students are required to grasp the meaning of numbers
(their values), the number symbolism, and the
relationship among numbers in a number system.
Meantime, in the aspect of number operations,
students should have a mental notion of the meaning
of operations, the impact of the operations, and the
relationship among the operations. Meanwhile, in the
aspect of calculation, students who have a good sense
of number can emerge various counting strategies
which based on their understanding of the numbers
and operations and reasonably select one of those
counting strategies that is considered more effective,
efficient and practical in solving computational
problems.
It is identified that not all researchers obviously
relate the notion of number sense to the ability to do
mathematical judgments, such as making estimation
or identifying error in calculation. Only NCTM
(2000) and Ali (2014) stress clearly that the ability in
doing mathematical judgments is a part of number
sense. NCTM (2000) describes the notion as ability
in making reasonable estimation, meanwhile Ali
(2014) states it as the ability in using benchmarks to
make mathematical judgments and recognize
unreasonable answers.
Although not stated obviously, some researchers
incorporated the mathematics judgment into the
ability in orchestrating numbers flexibly once dealing
with number-related problems. Fosnot & Dolk
(2001) and Putrawangsa & Hasanah (2018), for
example, incorporate the notion once students can see
various strategies in dealing with number-related
problems and intelligently decide one strategy among
the emergent strategies that are considered to be the
most efficient and fit with the problems. The selection
of the best strategy requires the students to use their
mathematical judgment by considering their
understanding of numbers, operations and the context
of the problems.
Although the researchers regard the notion of
number sense in many different ways, they all
consider the notion of number sense a comprehensive
faculty in dealing with numbers and number
operations in computational situation or problems
(see Table 1).
To enhance students’ sense of numbers they
should have: (1) an appropriate understanding of
numbers and how they are related one to another, (2)
an understanding the meaning of operations and how
they are related and impact numbers or other
operations, (3) a capability performing computation
by utilizing their understanding of the numbers and
the operations fluently and flexibly in dealing with
number-related problems including making
reasonable estimation, and (4) a faculty in making
appropriate judgment of calculation including
identifying potential error of a computation and
making estimation.
It implies that assessing students’ number sense
means assessing students’ understanding on the four
aspects, such as: (1) assessing students’ conceptual
understanding about numbers and how they are
related one to another, (2) assessing students’
conceptual understanding about operations and how
they are related and impact numbers or other
operations, and (3) assessing students’ proficiency in
performing calculation correctly and flexibly in
dealing with computational problems, including
making reasonable estimation, and (4) assessing
students’ faculty in making appropriate judgment of
calculation including identifying potential error of a
computation and making estimation.
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193

Table 1: Experts’ different views toward the elements of number sense.
NCTM (2000)
Fosnot & Dolk (2001)
Andrews & Sayers (2014)
Students who own number
sense, they:
1. Understand numbers (the
meaning of numbers);
2. Understand ways to
representing numbers;
3. Understand the relationship
between two numbers or
more;
4. Understand the number
system;
5. Understand the meaning of
numbers operations;
6. Understand the relationship
between operations;
7. Afford to compute fluently;
8. Afford to make a reasonable
estimation.
Three component constructing
students’ number sense:
1. Sense of Number
They grasp the idea of one-to-
one correspondence once
counting, the cardinality and
the magnitude of numbers,
hierarchical inclusion,
part/whole relationship, and the
notion of compensation.
2. Number System
They comprehend with the
idea of unitizing and place
value system.
3. Addition and Subtraction Facts
During computation, they are
proficient in utilizing ten-based
combination, commutative
property, doubling, swapping,
constant difference, and
canceling out.
Elements of students’ foundational
number sense:
1. Involving number cognition, its
vocabulary and its meaning;
2. Incorporating systemic
counting, including the notion
of ordinarily and cardinality;
3. Aware of the relationships
between numbers and quantity
(a number represent a certain
quantity);
4. Aware of the magnitudes of
numbers and the comparisons
between those magnitudes.
5. Aware and making estimation
of numbers;
6. Being able to perform simple
arithmetic operations, such as
addition and subtraction.
7. Aware of number patterns.
Ali (2014)
Jordan, Fuchs, & Dyson (2014) &
Jordan & Dyson (2016)
Putrawangsa & Hasanah (2018)
Students who own appropriate
number sense are proficient in:
1. Using numbers in flexible
ways when adding,
subtracting, multiplying or
dividing;
2. Using benchmarks to make
mathematical judgments;
3. Making mental calculations
and reasonable estimations;
4. Making predictions;
5. Understanding numerical
relationships between
mathematical concepts,
facts and skills; and
6. Recognizing unreasonable
answers.
Early development of number
sense involving
1. Mastering the notions of
numbers, such as the idea of
subitization, cardinality, and
the number symbolism.
2. Mastering the notion of
number relations, such as the
notion of the magnitude of
numbers, and the conceptual
structure of numbers (mental
structure of numbers).
3. Mastering early number
operation, such as
understanding the basic facts
of addition and subtraction
and being deft in
decomposing numbers in
various forms.
Students’ number sense comprises:
1. Understanding of numbers, such
as grasping the meaning of
numbers (their values), the
number symbolism, and the
relationship among numbers in a
number system.
2.Understanding of number
operations, such as understanding
the meaning of basic operations,
the impact of the operations, and
the relationship among the
operations.
3.Fluency in calculation, including
being able to emerge various
counting strategies and selecting
one of those counting strategies
that is considered more effective,
efficient and practical in solving
the computational problems.
3.4 Considerations in Assessing
Number Sense
Number sense is developed through multilevel of
cognitive development which is in line with the
human cognitive development of numbers and
operations. In the human cognitive development of
numbers, the natural number system is the basis for
more other advance number systems such as integers,
fractions, real, and complex numbers. Meanwhile,
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addition is regarded as the basis for other
mathematical operations, such as subtraction is
considered as the inverse mathematical operation of
addition, and multiplication is defined as repeated
additions. More advanced mathematical operations
are than developed from those basic mathematical
operations, such as division and exponentiation.
Regarding the multilevel of human cognitive
development of numbers and operations, students’
number senses are also developed according to such
a development. For instance, making sense of
discrete numbers is considered easier for students to
grasp comparing with making sense of numbers in
fractions. Having a sense of a mathematical
operation, such as addition, is required a lower
cognitive load comparing with other mathematical
operations, such as multiplication or division.
Therefore, making sense of addition is simpler to
conceive than multiplication or division. Here, it can
be concluded that students’ number sense is not a
static process, but it is a developmental process which
is developed over the time as the result of life
experiences, including educational experiences.
Hence, assessing students’ number sense requires
to consider the level of students’ cognitive
development about numbers and operations. It
implies that assessing early grade students’ number
sense may somewhat difference with those higher-
grade students regarding content and method of
assessment.
Regarding the multilevel of human cognitive
development of numbers and operations, it
exaggerates to claim that a single method or
instrument of number sense assessment is relevance
for the whole context of number sense assessment.
For example, an instrument for assessing early
graders’ number sense may be not sufficient to be
used to assess higher graders, or instrument of
assessment involving fractions will not be apt to be
used for students’ who have not learned yet about
fractions.
Moreover, the purpose of assessment leads to
several different approaches in assessing number
sense. At least, there are three different purposes for
assessing students’ number sense, such as screening,
diagnosing, and monitoring (Roberts, 2013).
Assessing number sense for screening purpose is
utilized once the assessment is administrated to
identify students’ risks in mathematics early or to
assess foundational core mathematics instructions.
Meanwhile, if the assessment intends to identify
students’ strengths or weaknesses, to develop
instructional interventions, or to categorize students,
the purpose of the assessment is to diagnostic
students’ number sense. The last, assessment for
monitoring is administrate donce the purpose is to
evaluate the efficacy of instructional interventions or
to take quick decisions regarding adjustment of
instructional interventions.
Hence, designing and developing instrument for
assessing students’ number sense regards at least
three considerations, such as (1) the level of students’
cognitive development (whether assessing pre-school
students, early grade students, high grade students,
etc.), (2) the mathematical scope and focus of the
assessment (whether focusing on a specific or holistic
operational skill, such as addition, subtraction,
multiplication, etc. on integers or fractions), and (3)
the purpose of the assessment (whether assessing for
screening, diagnosing, or monitoring).
4 CONCLUSIONS
The way the concepts of number sense perceiving
among scientists is slightly different. While cognitive
psychologists consider number sense as the ability to
deal with non-symbolic ideas of numbers, educators
tend to regard number sense as the faculty in dealing
with both symbolic and non-symbolic ideas of
numbers. However, both psychologists and educators
agree that students’ number sense, both symbolic and
non-symbolic ideas, influence students’ development
of advanced concepts of mathematics and students’
opportunity in the future career.
Regarding the impact of students’ sense of
numbers, assessing students’ number sense is crucial
to identify students’ early risk in mathematics and to
prevent them from failure both in advance
mathematics and in their future career.
Assessing students’ number sense means
assessing students’ understanding on the four aspects,
such as (1) assessing students’ conceptual
understanding about numbers and how they are
related one to another, (2) assessing students’
conceptual understanding about operations and how
they are related and impact numbers or other
operations, and (3) assessing students’ proficiency in
performing calculation correctly and flexibly in
dealing with computational problems, including
making reasonable estimation, and (4) assessing
students’ faculty in making appropriate judgment of
calculation including identifying potential error of a
computation and making estimation.
There are at least three points to be considered
once designing and developing instrument for
assessing students’ number sense, such as (1) the
level of students’ cognitive development (whether
Assessing Students’ Number Sense: What to be considered?
195

assessing pre-school students, early grade students,
high grade students, etc.), (2) the mathematical scope
and focus of the assessment (whether focusing on a
specific or holistic operational skill, such as addition,
subtraction, multiplication, etc. on integers or
fractions), and (3) the purpose of the assessment
(whether assessing for screening, diagnosing, or
monitoring).
ACKNOWLEDGMENT
Thanks to LP2M Universitas Islam Negeri Mataram
that has provided financial support for the current
research through DIPA UIN Mataram 2018.
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