Learning Number in Early Childhood and Its Relationship

with Mathematics Anxiety

Surya Sari Faradiba

, Cholis Sa’dijah

, I. Nengah Parta

and Swasono Rahardjo

Universitas Negeri Malang, Jalan Surabaya 6 Malang,Indonesia

Keywords: Number, Operations, Developmental Progression, Early Childhood, Mathematics Anxiety

Abstract : Numbers and operations are basic mathematics materials that can be taught to young children. This material

is very important because it is the basis for the development of the next mathematical abilities. Therefore,

we need to developmental progression in this topic, so as to provide a strong mathematical foundation for

children. A convenience sample of 20students was recruited from an elementary school in Malang,

Indonesia. Out of 20 students, only 2 of them were chosen as the sample under the consideration that they

are the most communicative.The interviews were semi-structured, covering main themes such as the number

and their operations, with follow-up questions to ensure that the interviewer had interpreted the answers as

intended.The tests were administered to students as individual interviews. The instrumentsused in this study

weremathematics tasks and the mathematics anxiety questionnaire consistingof 20 items with five response

choices in the form of face images.Students' answers to the mathematics task are analyzed based on

developmental progressionin learning number and operations, that is:Number-after equals one more,

Mentalcomparisons of close numbers, Number-after knowledge, Counting-based comparisons of collections

larger than or equal with four, Meaningful object counting, and Subitizing (small-number recognition). The

results of this study indicate that students who are incomplete in developmental progression have higher

math anxiety scores while learning mathematics

1 INTRODUCTION

According to Regulation Minister of Education and

Culture Republic of Indonesia Number 84, 2014

about early childhood education, Early Childhood

Education an effort to educate newly born to six

years children, is conducted by giving educational

stimuli to help children grow & develop physically

and spiritually so that the child has the readiness to

enter further education. Therefore, mathematics

learning at this stage needs special attention to

prevent mathematics anxiety.It is because, anxiety

might uppear during the childhood period, when

change occurs rapidly when negative attitudes and

anxiety occur at this age, it is usually persistent and

difficult to change even when they are adult. There

are some bad impacts of mathematics anxiety, that is

math avoidance (Hembree, 1990), distress (Tobias,

1978; Buxton, 1981) and interference with

conceptual thinking and memory processes (Skemp,

1986). Even for children, there appears to be a

negative correlation between anxiety and

achievement in mathematics (Hembree, 1990).

Although this correlation may be indirect, it is often

condisered that high levels of mathematics anxiety

impair performance.

Many studies show the relationship between

being good in early mathematics and achievement at

the next school level (Claessens, et al, 2009; Geary

et al, 2013; Sarama et al, 2012; Watts et al, 2014). In

fact, early math skills became the strongest predictor

of reading ability (Geary et al, 2013; Sarama et al,

2012; Watts et al, 2014) and mathematics (Clements,

et al, 2014; Duncan et al 2011, Bernsteinet al,

2014).Unluckly, children with low skill in math

have the tendency to be low achievers at the

education level (Watts, et al, 2014; Duncan, et al,

2011; Siegler, et al, 2012)

Early mathematics refers to a variety of basic

mathematical concepts for early childhood such as

calculating in rounded nineteen; quantity (more,

less, equal), shape (circle, square, triangle,

rectangle); spatial relationships (more than, below);

measurement (height, short, large, small, heavy,

light); and patterns, both in the form of color

patterns such as red, green, red, green, and image

patterns (National Research Council, 2009;

Faradiba, S., Sa’dijah, C., Parta, I. and Rahardjo, S.

Learning Number in Early Childhood and Its Relationship with Mathematics Anxiety.

DOI: 10.5220/0009916307530760

In Proceedings of the 1st International Conference on Recent Innovations (ICRI 2018), pages 753-760

ISBN: 978-989-758-458-9

753

Sarama& Clements, 2009). In natural way such

concepts of math will be explored by the children

during their interaction with the environment

(Sarama& Clements, 2009). For instance, when

small children build beam towers, they learn

mathematics by sorting the beams based on the size,

color, or shape. This is very concerned about spatial

relationships. In addition, these activities can also

develop reasoning skills. In this case, preschoolers

calculate or compare objects as they play, and find

out similarities and differences in patterns and

shapes (Seo and Ginsburg, 2004).

The general concern is that supporting early

mathematics might mean taking away time from

initial literacy. However, this does not have to be a

problem. The development of initial mathematical

skills and initial literacy are interrelated (IOM &

NRC, 2015) and attempts to support both can occur

simultaneously. In fact, when children learn

mathematics along with other subjects they learn

more math than if mathematics is taught in isolated

way (NRC, 2009). Children learn mathematics and

language in similar developments. Starting from

infancy, language skills and literacy development

with time when children build vocabulary, sentence

length, and complexity of their sentences. Children

learn how to building vocabulary, grammar mastery,

and ability to produce longer and more complex

sentences are the ways children use to learn how to

express their ideas in words (Kipping et al, 2012).It

is the same as the development phase in learning

early mathematics. Firstly, they learn the basic terms

in math, recognize the mathematics in tr surrounding

world, and learn to use more complex mathematical

concepts such as measurement, geometry, and

reasoning (IOM & NRC 2015; Janzen, 2008),

reading books, telling stories, and using "talking

math" are an easy andeffective way to engage and

improve math skills and early literacy development.

Many books for children highlight mathematics in

my ways.The purpose of this study was to describe

the utterances of childhood when they are learning

numbers and operations based on their

developmental progression.

2 METHOD

A convenience sample of 20 children was recruited

from an elementary school in Malang, Indonesia.We

used participants from one school to ensure that

students come from the samelearning environment.

All participants returned signed consent forms and

completed all interview assessment tasks.

Researchers chose to work with first graders

because students ages 6–8 typically function at the

uni-dimensional level (Case, 1996; Okamoto &

Case, 1996). Students are considered at the uni-

dimensional level because they are only able to use a

single mental number line to solve numerical

questions involving one dimension, such as adding

ones. At the uni-dimensional level, they can

determine that larger quantities, correspond to more

motor movements in the tagging process (which are

eventually done mentally and then not at all), to

number names further up in the naming sequence,

and to larger numerals, which are written further to

the right on a number line. By counting up and down

their mental number line, students can solve single-

digit addition and subtraction problems, where

addition (represented by the plus sign) results in

more and subtraction (represented by the subtraction

sign) results in less, and make judgments about the

relative magnitudes of two single-digit numbers

(Case, 1996; Griffin et al., 1995; Siegler&Ramani,

2008, 2009)

Out of 20 students, only 2 of them were chosen

as the sample under the consideration that they are

the most communicative. The interviews were semi-

structured, covering main themes such as the

number and their operations, with follow-up

questions to ensure that the interviewer had

interpreted the answers as intended. The interviews

ranged between 20 and 30 minutes. All interviews

were audio-taped and transcribed verbatim by the

researcher. In this study, researchers served as

interviewers.

The instrument in this study are mathematics

tasks and the mathematics anxiety questionnaire

contains 20 items with five response choices in the

form of face images.Students' answers to the task are

analyzed based on developmental progression in

learning number and operations, that is: (L1)

Number-after equals one more, (L2) Mental

comparisons of close numbers, (L3) Number-after

knowledge, (L4) Counting-based comparisons of

collections larger than or equal with four, (L5)

Meaningful object counting, and (L6) Subitizing

(small-number recognition).

The tests were administered to students as

individual interviews. Each studentwas interviewed

outside of the classroom in a nearby multipurpose

room using aset protocol. The student sat at a table

facing the wall, and the interviewer sat nextto the

student. We positioned a camera to capture the

student’s gestures and work and placed an additional

voice recorder near the student to pick up any

audionot captured on the camera. After the student

solved a problem, the interviewer asked, “How did

you solve it?” to learn more about his/her strategies

(Siegler, 1996). We did not give feedback on

correctness but provided generic encouragement

(e.g., “Great!”). For warm up, researcher ask about

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754

operations, like “What is addition?” If child is

unsure, then ask “What does it mean to add?” or

“Whathappens when you add?” Next, the researcher

asked about the subtraction, “What is

subtraction?”If the child is unsure, then ask“What

does it mean to subtract?” or “Whathappens when

you subtract?”

Table 1

Interview Question

Task Questions

Counting

backward

(Griffin,

2004)

Start at 10 and count backward as far as

you can.

If the child stops at 1: Can you count

backward any further? Is there anything

less than 1?

If the child stops at 0: Can you count

backward any further? Is there anything

less than zero? (Student answers) How

do you know?

If the child stops at a positive number:

Can you go any further?

If the child indicates the numbers keep

going: What would the last number be?

The counting task probed students’ knowledge of

thedecreasing verbal naming sequence. In this study,

students were asked to start at “10” and count

backwardas far as they could. The ordering task

provided additional information on how

studentsinterpret and order integers. This task also

probed how students coordinated their integer value

judgments with integer order because they then

indicated which numbers were the least and greatest.

In this study, the math anxiety questionnaire of

SEMA Questionnaire was translated into Indonesian

(Wu, et al, 2012). The first tenquestions related to

the mathematics curriculum and were used to

measure the anxiety related to math problem

solving. The other 10 questions were to assess the

anxiety related to social and testing situations

frequently faced the children while learning

mathematics.

Two students were administered the measure

individually in a one-on-one setting with an

interviewer. All questions on the SEMA was

presented on a piece of paper and also

simultaneously read aloud by the interviewer. After

each question, the children were asked to rate how

anxious they felt. Ratings were made on a five-point

faces response. Ratings were shown with graded

anxious and non-anxious faces in order to assist the

children in identifying their anxiety levels. Children

answered by selecting one of the faces. The tester

recorded the child’s answer, making sure to ask for

clarification if there was any obscurity. An

individual’s SEMA score was computed by

summing the all items’ ratings.

Figure 1.Pictorial Rating Scale

Source: Krinzinger et al. (2007)

Note: The type of questions are as follows: “How worried

are you if you have problems with … ?”

(veryworried to very relaxed). In this case, worried and

relaxed faces representan anxiety.

The point of this study is an interest in how the

students talk about number and operations. The

students’ verbally expressed experiences of number

and operations. The analysis begins by examining

how the students answer the questions about number

and operations. Answers in which the students

express their understanding were coded based on the

stages of the developmental progression (Frye,

2013). Researchers explore whether six stages of

specific developmental progression for number

knowledge are complete or not. In this case,

developmental progression is said to be incomplete

when there are students who are at a stage without

going through the previous stage. Furthermore,

researchers found a relationship with mathematical

anxiety caused by the stages that were missed in

developmental progression. Due to the small-scale

design, there will be no claims for generalization of

the results. Data included students’ written answers

as well as transcripts of any verbal information they

provided when solving themath problems.

3 RESULTS AND DISCUSSION

Initial experience with numbers and operations is

very important to obtain more complex

mathematical concepts and skills (NRC, 2009). This

learning process depends on identifying the

knowledge that children have and building that

knowledge to help them take the next step.

Developmental progression can help identify the

next step by providing teachers with indicators that

are appropriate for development to learn different

skills. First, children can start a new step with a

small number before moving to a larger number with

the previous step (Ginsburg, et al, 1998). This can be

seen from the following discussions:

(Answer of S1)

Teacher : “I have an apple while Dani has two, who

has more apples?”

Learning Number in Early Childhood and Its Relationship with Mathematics Anxiety

755

S1 : “Dani”

Teacher : “Can you help me to buy apples so I can

have more apples than Dani?”

S1 : “Mmmm, yes I can”

Teacher :” how many apples do I need to buy

again?”

S1 : “an apple”

Teacher : “Oh yes, if I buy 1 apple, how many

apples do I have now?”

S1 : “1 plus 1 to 2”

Teacher : “is my apple more than Dani's?”

S1 : “I don't think so, the apples are both two”

Teacher : “good, so buying 1 apple isn't enough, I

need to buy 1 apple. So how many apples

do I have now?”

S1 : “2 + 1 = 3”

Teacher : “Good”

(Answer of S2)

Teacher : “I have an apple, while Dani has two.

Who's more?”

S2 : “Dani”.

Teacher : “How do you know?”

S2 : “2 apples are the same as an apple we

bought twice”

S1 and S2 can give the correct answer, that is

more Dani’s Apples. In other words, S1 and S2 are

able to compare which of the two numbers (1 and 2)

are more. So that you can dismiss S1 and S2 in the

first stage of developmental progression, namely

number-after equals one more. This is not surprising

because we know that children think mathematic

matters in early age even before going to school.

Children show their understanding of different

concepts of numeracy informally before they began

to learn math at school (Song, et al, 1987; Sophian,

et al, 1995). This is informal numeracy knowledge,

that is, skills that children develop before starting

school that does not depend on written mathematical

notation (Purpura& Napoli, 2015). In this study,

both S1 and S2’s knowledge of numeracy indicates

their counting skills development. It also proves the

capacity to compare, share, order, estimate and

calculate different quantities. In infancy, and innate

basic skills in recognizing and responding to

numerical cues and apparent (Wynn, 1995;Xu, et al,

2005).

Knowing what is 'more' and 'less' means to help

children recognize the way to compare numbers. It is

really important for the math developmental since

children will be able to compare groups of objects

and calculate them later. In this study, the important

thing is that after children can mentally compare

numbers and see that "2" is one more than "1" and

that "3" is one more than "2." They can conclude

that any number in the order of calculation is exactly

one time more than the previous number. A child is

ready for the next step when he recognizes, for

example, that "five" is one more than "4." This can

be seen from the following discussion:

(Answer of S1)

Teacher :”3 and 2, which one is more?”

S1 : “3”

Teacher : How do you know?

S1 : “because in the number line, 2is written

before number 3”

Teacher :“ If I write number 3 before number 2,

what do you think? is it still 3 greater

than 2?”

S1 : “no, 2 is greater than 3”

(Answer of S2)

Teacher : “3 and 2, which one is more?”

S2 : “3”

Teacher : “How do you know?”

S2 : “Because 3 sequences are after 2”

Teacher : “How far is 3 to 2?”

S2 : “Not too far away, just one jump

(laughing), I mean between 2 and 3

there are no other numbers”

In the conversations above we can see that the S2

has passed the second stage (L2) and the third stage

(L3) of the developmental progression. It is shown

when S2 says numbers 2 and 3 are sequential, there

are no other integers between them. Besides that, it

is also indicated by the answer S2 which represents

2 apples equal to buying 1 apple 2

times.Meanwhile,S1 failed in the L2 and L3 stages.

It appears that S1 is fixed on the position of numbers

in the number line, and argues that the more to the

right the number gets bigger, by forgetting the

condition that the number must be written in

sequence. Griffin (2004) has proposedcertain

number sense content for the typical five-year-old

child. According to her, knowing numbers in the

counting sequence have a fixed position.

Once children recognize that counting can be

used to compare collections and have the number-

after knowledge, they can efficiently and mentally

determine the larger of two adjacent or close

numbers (e.g., that “9” is larger than “8”). A child

has this knowledge when he or she can answer

questions such as, “Which is more, seven or eight?”

and can make comparisons of other close

numbers.Once children can compare collections and

have number-after knowledge, they can efficiently

and mentally determine which is greater than 2

adjacent or near numbers (for example, that "9" is

greater than "8"). A child has this knowledge when

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he can answer questions like, "Which is more, seven

or eight?" And can make comparisons of other close

numbers.

Familiarity with the order of counting allows a

child to have number-after knowledge. For example,

to enter a sequence at any time and specify the next

number, not always counting from one. A child is

ready for the next step when he can answer

questions like, "What happens after 3?" By stating

"3, 4" or just "4" instead of, say, counting "1, 2, 3, 4"

At this stage, only S2 can enter this stage. While S1

cannot yet, so S1 still needs a better understanding

of the number concept in the second stage.

After children can use small number recognition

to compare small collections, they can use

meaningful object counting to determine the larger

of the two collections (for example, "7" items are

more than "6" items because you have to count

further). A child is ready for the next level when

they shown two different collections. This can be

seen from the following discussion.

(Answei of S1)

Teacher :“9 books and 6 books, which one

is more?”

S2 : “9 books is more”

Teacher : “How do you know?”

S2 : “Because when we mention numbers

1,2,3,4,5,6,7,8,9 we must call 6 first

rather than 9 so 9 more than 6”

(Answer of S2)

Teacher :“9 books and 6 books, which one

is more?”

S2 : “9 books is more”

Teacher : “How do you know?”

S2 : “yes because 9 means 9 steps from number

0 while 6 means 6 steps from number 0,

so that 9 steps are farther than 6, so 9 is

more than 6”

The above conversation indicates that S1 and S2

are at stage (L4) Counting-based comparisons of

collections larger than or equal with four and (L5)

Meaningful object counting. In this case, meaningful

object counting is counting in a one-to-one

correspondence and recognizing that the last word

used while counting is the same as the total (this is

called the cardinality principle).

(Answer of S1)

Teacher : (given 4 blocks) “How many?”

S2 : (she counts by pointing and assigning one

number to each block) “1,2,3,4 …. So

that the total is 4.”

Teacher :” Next, can you count 4 + 2 = … ?”

S2 : “Wait, 1,2,3,4,5,6. So, the answer is

6“.

(Answer of S2)

Teacher : (given 4 blocks) “How many?”

S2 : (she counts by pointing and assigning one

number to each block) “1, 2, 3, 4…. So

that the total is 4”

Teacher :“Next, can you count 4 + 2 = … ?”

S2 : “Of course, 4,5,6. So, the answer is

6”.

Aimprovement of the counting-all procedure is

the counting-on procedure.In the counting-all

procedure, the sum is found by counting the total

number of entities which comprise the addeds. So in

finding the sum of 4 and 2, S1 verbalized,

“1,2,3,4,5,6”. The counting-on procedure, however,

is more sophisticated. Here the child begins the

count with the number name whichrepresent the

numerosity of one of the addeds. So in finding the

sum of 4 and 2, S2 would verbalize, “4,5,6”. While

it is more effective to begin with the larger of the

two addeds, and while indeed children can often be

observed using this more effective procedure, others

will begin the counting-on procedure from the first

added, even if it is smaller.

Furthermore, S2 understanding of cardinality

(the number of elements in a set or other grouping as

property of that grouping) was found to be

infrequently examined. However, with increasing

age, children tend to spontaneously emphasize and

repeat the last word in a count sequence

(Cordes&Gelllman, 2005).

Subitization is the ability of young children to

immediately recognize the total number of items in a

collection. The next step is give the label them with

the appropriate number words. When children are

presented with many examples of different

quantities (like 2 eyes, 2 socks, and 2 books)

labeled with the same number word, not examples

that are labeled with other numeric words (three

pencils), children are in the process of making the

right concept.The most frequent subitization that

occurs in children is subitization of numbers 1, 5 and

10. This is because the two numbers are very

familiar in the child's memory. Both, S1 and S2

represent one number by showing one finger (S1 and

S2 use the index finger), and bends the other four

fingers. Meanwhile, the representation of the

number five is shown by S1 and S2 by opening his

palm wide, where the five fingers are upright. The

same thing is done by S1 and S2 when representing

10, namely by opening both of his palms and lifting

all his fingers. Meanwhile the subitization process

for other numbers also appears on S1 and S2

differently. This can be seen when S1 and S2

Learning Number in Early Childhood and Its Relationship with Mathematics Anxiety

757

represent number 3, that is, by raising the three

fingers on the right hand in the sequence, which are

index finger, middle finger, and ring finger. The

same thing also happens when S1 and S2 represent

number 4, the fingers used are the four fingers of the

right hand which is sequentially located, the little

finger, ring finger, middle finger, and index finger.

In this section, we can conclude that the S1

developmental progression experienced by S1 is

incomplete. Based on the results of the interview, we

know that S1 only controls L1, L4, L5, L6. S1 failed

in the L2 and L3 stages. It appears that the S1 is

fixed on the number line, and the arguments that are

the right number will be bigger, by forgetting the

conditions that must be written in sequences.

Meanwhile, S2 can do all the stages in

developmental progression (L1-L6) completely

Based on the results of the interview, we know

that developmental progression from S1 is

incomplete, while S2 is complete. Furthermore, to

explore their knowledge about numbers, researchers

present the counting backwardtask(Griffin, 2004).

(Answer of S1)

Teacher: “Start at 10 and count backward as far as

you can!”

S1 : “10, 9, 8, 7, 6, 5, 4, 3, 2, 1”

Teacher: “Can you count backward any further? Is

there anything less than 1?”

S1 : “I don’t know”

(Answer of S2)

Teacher: “Start at 10 and count backward as far as

you can!”

S2 : “10, 9, 8, 7, 6, 5, 4, 3, 2, 1”

Teacher: “Can you count backward any further? Is

there anything less than 1?”

S2 : “mmm, zero maybe”

Teacher: “How do you know?”

S2 :”One means that there is something as

much as one thing, but zero means that

there is no object at all”

In general, S1 and S2 can complete the backward

count task properly. Even though S1 still doesn't

understand numbers smaller than 1.While, S2 uses

cardinality to argue about 0 which is smaller than 1.

Because S2 can answer questions correctly, in this

case it can be concluded that S2 has a mathematical

performance in accordance with its developmental

progression.

S1 has a mathematics anxiety score 60 of the

maximum total score is 80 with details of 43 points

coming from the first part of the questionnaire which

contains questions related to mathematics anxiety

related to academic. Meanwhile, the remaining 17

points are math anxiety in everyday life. So that S1

can be confirmed to have high math anxiety.

Meanwhile the S2 has an anxiety score of 21 of the

maximum total score of 80, with details of 11 points

coming from the first part of the questionnaire, and

10 points from the second part of the questionnaire.

So that it can be categorized as S2 having low

mathematical anxiety. It might be that for primary

school students the correlation between math anxiety

and calculation performance is weak to nonexistent,

for example because math anxiety may be more

related to personality aspects such as general anxiety

or because it might be very strongly mediated by

teachers' or parents' attitudes (Stevenson et al.,

2000).

4 CONCLUSION

Children demonstrate an interest in math well in

early year of school. There are six stages in learn

number and operations using developmental

progression in early childhood, that is: (L1)

Number-after equals one more, (L2) Mental

comparisons of close or neighboring numbers, (L3)

Number-after knowledge, (L4) Counting-based

comparisons of collections larger than or equal with

four, (L5) Meaningful object counting, L(6)

Subitizing (small-number recognition). The first

stage and the second stage can occur simultaneously,

and allow for no sequential. Counting sequence

understanding helps children to cope withmore

complex problems. With good development of

counting sense as a problem solving tool, children

may use counting to compare size off two set count

objects accurately without a need to touch the

objects physically, use counting to solve problem of

simple addition and subtraction and apply more

complex counting strategies such as counting on

from the larger set.

As a result of this study we know that S1 has

incomplete developmental progression while a

complete S2. But both can complete counting

backward tasks well. However, math anxiety scores

on S1 are much higher than S2. This means that in

teaching mathematics to early childhood we cannot

use orientation on academic results. But, focus on

the child's process in understanding each concept in

mathematics.

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