Investigation of primary school students solving arithmetic verbal problems

Journal of Education and Learning (EduLearn)
Vol. 16, No. 1, February 2022, pp. 76~84
ISSN: 2089-9823 DOI: 10.11591/edulearn.v16i1.20336  76
Journal homepage: http://edulearn.intelektual.org
Investigation of primary school students solving arithmetic
verbal problems
Ayten Pınar Bal, Perihan Dinç Artut
Mathematics and Science Education Department, Education Faculty, Cukurova University, Adana, Turkey
Article Info ABSTRACT
Article history:
Received Jul 9, 2021
Revised Nov 19, 2021
Accepted Jan 15, 2022
The study determined the arithmetic verbal problem solving skills of primary
school students and to examine them according to grade level and
mathematics achievement variables. The study was designed according to
the relational survey model, one of the quantitative research methods. The
population of the research consists of students studying in public primary
schools in the central districts of a large city in the south of Turkey. The
sample consisted of 1,865 students determined according to the
disproportionate stratified sampling method. The "arithmetic verbal problem
test for students" prepared by the researchers was used as a data collection
tool. As a result of the research, it was revealed that the students were most
successful in the problems of combining and separating with “consequence
unknown”, then “with change unknown” and “initial unknown”, but they
were less successful in part-whole type problems. Thus, it was concluded
that as the mathematics achievement and grade levels of the students
increased, their success in solving verbal problems also increased.
Keywords:
Arithmetic verbal problems
Mathematics lesson
Primary school students
Problem structures
Problem test form
This is an open access article under the CC BY-SA license.
Corresponding Author:
Ayten Pınar Bal
Mathematics and Science Education Department, Education Faculty, Cukurova University
Balcalı, Çukurova Üniversitesi Rektörlüğü, 01330 Sarıçam/Adana, Turkey
Email: apinar@cu.edu.tr
1. INTRODUCTION
Mathematics is one of the most important skills that a person must master in order to establish
relationships between school practices and daily life practices, to manage life activities well, and to achieve
economic success [1]–[4]. This study focuses on problem solving, which is at the center of mathematics
teaching. Problems are difficult situations that the person has not encountered before and is unprepared for a
solution. Dewey [5] defined the problem as anything that confuses the human mind, challenges it, and
obscures belief. Kurlick and Rudnick [6] defined the problem as a quantitative or other situation in which the
person cannot see or find a clear way to solve it. According to Bransford and Stein [7], the problem consists
of inconsistencies between the initial state and the target state. When the definitions related to the problem
are examined in general; situations that are new for the person, challenging the person, arousing the curiosity,
and requiring the use of rules specific to a mathematics in the solution are discussed [8].
Problem solving is a rather vague concept that is thought of as an umbrella where different
theoretical approaches coexist [9]. According to mathematicians, problem solving occurs when the algorithm
to be applied is not obvious. Polya [10] identified some strategies for problem solving. In the first stage of
problem solving, it is necessary to understand the problem and to see clearly what is necessary. In the second
stage, determining the relationship between what is given in the problem and what is wanted and making a
plan for the solution; implement the plan in the third phase; stated that it is necessary to reconsider the
solution obtained in the fourth stage [10].

J Edu & Learn ISSN: 2089-9823 
Investigation of primary school students solving arithmetic verbal problems (Ayten Pınar Bal)
77
Problem solving forms the center of mathematics programs [11]. During problem solving activities,
students can learn to solve a problem in a systematic way and to reveal how they think in this process. In
addition, students discover new ways of thinking by solving problems, and all this helps them feel confident
when faced with unfamiliar events in life. As their problem-solving skills improve, students' self-confidence
increases and they learn to respect and value the ideas of others [12]. With problem solving, mathematical
reasoning skills are developed, more meaningful learning is provided and strong bonds are formed between
mathematical concepts [13]. Knowing the types, contents and purposes of use of the problems that should or
have been dealt with in mathematics lessons is an important condition in the mathematics learning-teaching
process.
Problems were classified differently by different researchers at different times. When the different
problem classifications are considered collectively and primary school mathematics lessons are taken into
account, it can be said that verbal problems based on four operation skills are frequently included in
mathematics education, which aims to provide children with basic skills. Such problems are closed or routine
verbal problems that can be solved with the help of four operations and have a single result [14], [15]. They
are mathematical word problems based on four operations in elementary school 1-4 textbooks.
Mathematical word problems reflect the capacity and understanding of applying mathematical ideas
in daily life and in the service of science, technology, engineering and advanced mathematics [16]. Solving
word problems in mathematics often means extracting new information from given data. Known information
may be collected by the solvent itself or provided by various means. To describe a real-world situation, the
solver needs to select those aspects of the situation he or she wants to focus on and decide which objects
should be explicitly stated [9]. Problem solving tasks, in this context, arithmetic and verbal problem solving
tasks constitute an important part of the mathematics curriculum at primary school level. The most important
reason for using such problems in schools is to teach students to apply the mathematical knowledge and skills
they learned at school in the real world [4], [17]. On the other hand, problem solving opens a window to
children's mathematical thinking. It provides opportunities for them to understand mathematics [18]. Stigler
et al. [19] stated that studies have revealed that the semantic structure of the problem, the location of the
unknown quantity, and how the problem is expressed are effective in solving a particular problem. Children's
difficulties in solving verbal problems result from both developmental limitations and limited opportunities
for learning and practice.
When the literature on problem solving is examined, it is seen that researchers [20], [21]. Van de
Walle et al. [15] categorize arithmetic verbal problems in terms of the relationships they contain. According
to this classification, combining (subcategories; outcome unknown, change unknown, initial unknown),
separation (subcategories; outcome unknown, change unknown, initial unknown), comparison
(subcategories; difference unknown, small unknown, large unknown) and fragment- It was analyzed in four
main categories as part-whole (subcategories; whole unknown, part unknown).
In his book "How to Solve it", Polya [10] describes the act of problem solving as "Solving a big
problem is a big discovery, but there is a bit of discovery in the solution of every problem. Even if your
problem is not big, if it arouses your curiosity, offers you an opportunity to develop your creativity, and if
you can solve it based on your own knowledge, you can experience the tension and enjoy the victory. This
explanation reveals the importance he attaches to problem solving [10].
Curriculum changes that encourage students' in-depth understanding and emphasize the importance
of thinking emphasize that problem solving is an important mechanism for developing these skills. However,
problem solving is not a new curriculum or pedagogical innovation. More than 60 years ago, the importance
of problem solving was recognized and this importance was strongly emphasized throughout the 1980s [22].
While national council of teachers of mathematics (NCTM) [23] emphasized that problem solving
“should be central to the mathematics curriculum”. In addition, NCTM [24] curricula are available to all
students from pre-school to twelfth grade; It has been revealed that it is necessary to provide opportunities for
creating new mathematical knowledge through problem solving, solving problems arising from mathematics
and other content, applying and using the appropriate one of various strategies to solve problems, observing
and thinking deeply on mathematical problem solving processes. The idea that problem solving should be a
part of mathematics programs continued in the following years. In this context, NCTM [23] reiterated its call
in 2003 for problem solving to form an integral part of the mathematics curriculum. However, there is little
evidence to suggest that this focus on problem solving occurs in today's classrooms [25]–[27]. In this
direction, it is thought that it is important to conduct new studies on problem solving at all levels.
In addition, problem solving makes an important contribution to the development of 21st
century
learning skills. Countries take 21st
century learning skills into account when arranging their curricula. These
skills were taken into account when creating curricula in Turkey. In order for children to be successful in
solving verbal problem types, all verbal problem types should be adequately presented to children by teachers
[28]–[34]. However, studies show that teachers mostly include problem types with unknown results when
presenting verbal problems to children [28], [34], [35].

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When the literature on arithmetic verbal problems is examined [36]–[43] studies are mostly carried
out on preschool children. It has been observed that there are a limited number of studies at primary level
within the framework of available resources. Olkun and Toluk [44] applied arithmetic verbal problems based
on addition and subtraction on a total of 60 third and fourth grade students according to the relationships they
contain. In this study, it was revealed that students were more successful in solving problems with more
result unknowns. Tarım [34] applied arithmetic verbal problems based on addition, subtraction,
multiplication and division to third graders according to the relationships they contain. The result of this
research showed that more students are more successful in solving problems with result unknowns. Based on
this fact, it is thought that determining the arithmetic verbal problem-solving skills of students studying at all
levels of primary education based on addition, subtraction, multiplication and division will improve both
their processing skills and their problem-solving skills. However, many studies on arithmetic verbal problems
have been found in the relevant literature. Accordingly, determining the existing situation of primary school
students on arithmetic-verbal problems is very important in terms of both classroom teachers, the organizers
of the textbooks, and the experts who develop the education program in terms of their knowledge. In this
context, answers to the following questions were sought: i) What are primary school students' achievement
scores for arithmetic-verbal problems? ii) Is there a significant difference between grade levels and primary
school students' scores in solving arithmetic verbal problems? iii) Is there a significant difference between the
mathematics achievement levels of primary school students and the scores they get from arithmetic verbal
problems?
2. RESEARCH METHOD
This study aimed to determine the arithmetic verbal problem solving skills of students attending all
levels of primary school and to examine them according to some variables, was designed according to the
relational survey model, one of the quantitative research methods. It is a survey that reveals the change
between at least two variables according to the relational survey model. In this model, it aims to determine
whether the variables examined change together and if there is a change [45]. In this study, it is tried to
determine the change process between the arithmetic verbal problem solving skills of primary school students
and their grade level and mathematics achievement.
2.1. Population and sample
The population of the research consists of students studying in public primary schools in the central
districts of a large city in the south of Turkey. The sample consists of a total of 1,865 students determined
according to the disproportionate stratified sampling method. In the disproportionate stratified sampling
method, before selecting the sample, the universe is divided into various strata and random samples are
selected from the divided plates [46]. In other words, in the disproportionate stratified sampling process, only
the number of strata is considered and the proportions of these strata within the universe are not taken into
account. Within the scope of this study, the universe is divided into three layers according to the socio-
economic status of the regions where the schools are located. The number of students studying in these
schools varies by region. Accordingly, the student information included in the study is given in Table 1.
Table 1. Distribution of personal characteristics of the students in the sample group
Personal traits Variables F %
Gender
Female 925 49.6
Male 940 50.4
Total 1,865 100
Grade level
First class 466 23.6
Second class 480 25.7
Third class 462 25.9
Fourth class 457 24.5
Total 1,865 100
Mathematics achievement levels
Should be improved 673 36.1
Good 668 35.8
Very good 524 28.1
Total 1,865 100
School's socio-economic level (SEL)
Lower SEL 689 38.0
Medium SEL 672 36.0
Upper SEL 504 27.0
Total 1,865 100

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Table 1 shows that the distribution of the students in the sample group according to the variables of
gender, grade level, mathematics achievement level and socio-economic level of the school is presented.
Accordingly, 49.6% of the students participating in the study were female and 50.4% were male students.
23.6% of the students attend the first grade, 25.7% attend the second grade, 25.9% attend the third grade and
24.5% attend the fourth grade. On the other hand, when the mathematics achievement levels are examined,
36.1% of the students should be improved, 35.8% are at the level of "good" and 28.1% are at the level of
"very good". In addition, when the socio-economic level of the schools where the students participating in the
sample are studied, 35.9% of the students are educated in schools in the lower SEL, 36% in the middle SEL
and 28% in the upper SEL.
2.2. Data collection tool
As a data collection tool in the study, a draft "Arithmetic Verbal Problem Test for Students" was
created by the researchers to determine the arithmetic verbal problem solving skills of primary school
students. Whether the problems in the draft form are suitable for primary school students were presented to
the opinions of five classroom teachers and two lecturers who are experts in mathematics education problem
solving. The test was given its final form in line with the feedback received from them. The prepared test was
arranged according to the classification of arithmetic verbal problems for addition-subtraction-multiplication-
division [15]. Accordingly, arithmetic verbal problems are classified as combining, separating, part-part-
whole and comparison problems. According to this classification, combining (subcategories; outcome
unknown, change unknown, initial unknown), separation (subcategories; outcome unknown, change
unknown, initial unknown), comparison (subcategories; difference unknown. small unknown. large
unknown) and part, part-whole (sub-categories; whole unknown, part unknown) were examined in four main
categories. The data collection tool consists of 11 questions, one from each category.
2.3. Data collection
School administrators and teachers were interviewed by contacting the schools where the
application would be made. In these interviews, the purpose of the study was explained and the necessary
permissions were obtained from the parents of the students and the implementation process was started. In
the application of the arithmetic verbal problem test for the students, the necessary information was given to
the classroom teachers and the forms were filled in one lesson. In addition, information about the general
success of the students in the mathematics lesson was obtained from the classroom teachers.
2.4. Analysis of data
The questions in the arithmetic verbal problem form for students were examined one by one by two
researchers. Students' solutions were evaluated as 1 point if they were correct, and 0 points if they were
incorrect or unanswered. In the analysis of the obtained data, descriptive statistics and multivariate analysis
of variance (MANOVA) analyzes were applied with the help of SPSS 22.0 program. Before the analysis of
the data, Shapiro Wilks, Box's M test and Levene test were performed on the data and it was concluded that
the data were suitable for the parametric test conditions [47].
3. RESULTS
This study conducted to determine the arithmetic-verbal problem solving skills of primary school
students. It examined the relationships between these skills and their grade level and mathematics
achievemen. Table 2 presents the distribution of students' scores in solving arithmetic-verbal problems.
Table 2. Distribution of primary school students' achievement scores for arithmetic verbal problems
Arithmetic verbal problem type Category f Mean Ss
Join Result unknown 1865 .90 .23
Change unknown 1865 .68 .41
Initial unknown 1865 .69 .41
Separation Result unknown 1865 .79 .35
Change unknown 1865 .69 .41
Initial unknown 1865 .60 .40
Comparison Difference unknown 1865 .61 .44
Little unknown 1865 .58 .44
Big unknown 1865 .55 .45
Part-whole All unknown 1865 .54 .47
Part unknown 1865 .40 .45

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Table 2 reveals that the students are successful in solving problems in the join. This is followed by
problems of separation (subtraction), comparison (multiplication) and part-whole (division). According to
Table 2, when the results related to the sub-categories of each problem type are examined, it is seen that the
combination type problems are successful in the problems with “consequence unknown”, and “change
unknown” and “initial unknown” (X=69). Type problems are seen to be less successful. On the other hand, it
is seen that students are most successful in separation-type problems with "result unknown" (X=79), then
"change unknown" (X=69) and "initial unknown" (X=60) types of problems. In addition, in comparison type
verbal problems, it was found that the students were close to each other and moderately successful in
"difference unknown" (X=61), "small unknown" (X=58) and "large unknown" (X=55) types of problems.
Finally, in part-whole type problems, students were less successful in "whole unknown" (X=54) and "part
unknown" (X=40) types of problems.
As the second sub-objective of the study, the scores obtained by the students in solving arithmetic
verbal problems according to their grade levels were analyzed and the results are given in Table 3 (see
Appendix). The table shows the distribution of students' achievement scores in solving verbal problems
according to grade levels. According to the grade level, the scores of the students in solving the verbal
problems show a significant difference in all categories except the big unknown category (F=1.48; p>0.05).
According to the results of the analysis applied to determine the direction of the difference, it is seen that the
students studying in the upper classes are more successful than other students. It can be said that as the grade
level of the students increases, their success in solving verbal problems increases. Based on to the last sub-
purpose of the study, the scores obtained by the students in solving arithmetic verbal problems were analyzed
referring to their mathematics achievement levels and the results are given in Table 4.
Table 4. Analysis results of students' scores in solving arithmetic verbal problems according to mathematics
achievement levels
Arithmetic verbal
problem type
Category
Mathematics
achievement levels
N X ss sd F p
Eta
square
Significant
difference
Join
Result
unknown
Should be improved 673 .82 .31
2 63.342 .00 0.65 3.2>1
Good 668 .94 .17
Very good 524 .95 .14
Change
unknown
2 237.957 .00 .20
3>1.2
2>1
Good 668 .78 .35
Very good 524 .87 .29
Initial
unknown
2 211.126 .00 .18
3>1.2
2>1
Good 668 .78 .36
Very good 524 .87 .28
Separation
Result
unknown
2 155.651 .00 .14
3>1.2
2>1
Good 668 .85 .28
Very good 524 .93 .20
Change
unknown
2 94.195 .00 .09
3>1
2>1
Good 668 .76 .36
Very good 524 .80 .35
Initial
unknown
2 45.667 .00 .04 3>1.2 2>1
Good 668 .62 .39
Very good 524 .71 .39
Comparison
Difference
unknown
2 50.128 .00 .05 3.2>1
Good 668 .60 .44
Very good 524 .65 .44
Little
unknown
2 37.467 .00 .03 3.2>1
Good 668 .64 .43
Very good 524 .66 .44
Big
unknown
2 10.969 .00 .01 3.2>1
Good 668 .65 .43
Very good 524 .65 .44
Part-whole
All
unknown
2 1.429 .24 .00
Good 668 .57 .46
Very good 524 .53 .48
Part
unknown
2 14.02 .00 .02 3.2>1
Good 668 .43 .46
Very good 524 .46 .47
Table 4 shows the distribution of students' achievement scores in solving verbal problems according
to their mathematics achievement levels. According to this, the scores that students get in solving verbal
problems according to their mathematics achievement levels show a significant difference in all categories
except for the "whole unknown" category (F=1.429; p>0.05) in the part-whole type. According to the results

81
of the analysis applied to determine the direction of the difference, it is seen that students with very good and
good success are more successful than students at the level of improvement. Accordingly, it can be said that
as students' mathematics achievement increases, their success in solving verbal problems also increases.
4. DISCUSSION
As a result of this study, which was carried out with the aim of determining the arithmetic verbal
problem solving skills of students attending all levels of primary school based on addition, subtraction,
multiplication and division and examining them according to some variables, it has been revealed that
students' success in combining and separating problems is higher than in comparison and part-whole
problems. One reason may be that the operations for comparison and part-whole problems are more complex
than joining and separation. In comparison problems, the operation structure also includes the addition
operation. Similarly, the operation structure of part-whole problems includes subtraction and multiplication
operations. It can be thought that this situation may have caused students to be less successful in solving
problems based on such operations [33], [44].
When each problem type is considered in terms of sub-categories, it has been seen that students are
most successful in combining and separating problems with "result unknown", then "change unknown" and
"initial unknown" problems. In addition to this, it was concluded that the students were successful at least in
"part unknown" (X=40) type problems in part-whole problems. When the relevant literature is examined, it is
seen that there are similar results to the results of this research [31], [32], [48], [49]. For example, in the
study of Sarıbaş and Aktaş-Arnas [32] in which they examined 57 mathematical activity books containing
verbal problems related to addition and subtraction, the majority of the activity books only include problem
examples with unknown results. Of these problems, 84.2% of them are combining-result unknowns and
78.9% separation-result unknowns. The reason why the students in this study achieved more success in the
sub-category of unknown results may be similar.
According to the second sub-goal of the study, it was concluded that as the grade levels of the
students increased, their success in solving verbal problems also increased. As students' grade levels increase,
naturally, their experience of mathematics and their knowledge and practices in this context also increase.
From this, it can be said that the grade level supports the positive increase in success in solving such
problems. On the other hand, it was concluded that as the mathematics achievement of the students increased,
their success in solving verbal problems also increased [50]–[53]. For example, in the study of Yeşilova [53],
it was concluded that students with good mathematics achievement had higher problem-solving success than
other students, they used different strategies more, and their solutions were more detailed and
understandable. From this point of view, it can be said that the results obtained from this study are in parallel
with the literature findings. Students who are more successful in mathematics can show a higher performance
in solving verbal problems than other students.
5. CONCLUSION
These results revealed that the students were most successful in the problems of combining and
separating with "consequence unknown", then in "change unknown" and "initial unknown" problems, but
they were less successful in part-whole type problems. On the other hand, it was concluded that as the
mathematics achievement and grade levels of the students increased, their success in solving verbal problems
also increased. When these results are taken together, not all verbal problems are included in the textbooks
with the same frequency. It can be thought that the fact that the sub-categories of each of these problem types
has different operation structures and complexity levels affects the success of students in solving these
problems.
In addition, it can be said that grade level and mathematics achievement also affect success in all
problem types discussed in this research. The study is limited only to primary school students' arithmetic
verbal problem solving skills related to addition and subtraction. In another study, classroom teachers or
mathematics teachers' in-class practices or awareness levels on verbal problem types can be examined.
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APPENDIX
Table 3. Analysis results of students' scores in solving arithmetic verbal problems according to grade levels
Arithmetic verbal
problem type
Category
Class
(n=466)
Class
(n=480)
Class
(n=462)
Class (n=457)
X ss X ss X ss X ss
Join
Result unknown .82 .31 .92 .21 .93 .17 .93 .20
Change unknown .47 .43 .58 .44 .82 .33 .85 .31
Initial unknown .46 .44 .63 .42 .84 .31 .81 .34
Separation
Result unknown .66 .41 .75 .37 .89 .26 .85 .292
Change unknown .48 .41 .65 .43 .78 .36 .82 .34
Initial unknown .52 .42 .56 .38 .62 .41 .69 .37
Comparison
Difference unknown .45 .44 .47 .46 .62 .44 .66 .43
Little unknown .52 .45 .52 .44 .61 .45 .68 .42
Big unknown .62 .42 .60 .45 .59 .44 .65 .43
Part-whole
All unknown .40 .45 .57 .45 .59 .47 .61 .46
Part unknown .28 .42 .42 .44 .41 .46 .50 .47
Table 3. Analysis results of students' scores in solving arithmetic verbal problems according to grade levels
(Continued)
Arithmetic verbal
problem type
Category sd F p Eta
square
Significant
difference
Join Result unknown 3 26.96 .00 .04 2>1
3.4>2
Change unknown 3 106.48 .00 .15 3.4>1.2
Initial unknown 3 94.0 .00 .13 4.3>1.2
2>1
Separation Result unknown 3 43.57 .00 .07 4.3.2>1
4>3.2
3>2
Change unknown 3 71.29 .00 .10 4.3>1.2
2>1
Initial unknown 3 16.58 .00 .03 4>1.2.3
3>1.2
2>1
Comparison Difference
unknown
3 26.08 .00 .04 4.3>1.2
3>1.2
2>1
Little unknown 3 13.91 .00 .02 4>1.2.3
3>1.2
Big unknown 3 1.48 .217 .00
Part-whole All unknown 3 19.69 .00 .03 4.3>2
2>1.
Part unknown 3 19.46 .00 .03 4>1.2.3
3>2
2>1

 ISSN: 2089-9823
84
BIOGRAPHIES OF AUTHORS
Ayten Pınar Bal is associate professor at the Education Faculty, Cukurova
University. Her current research focuses on mathematics education in teacher education,
teaching mathematics in elementary school, cooperative learning and problem solving. She
can be contacted at email: apinar@cu.edu.tr.
Perihan Dinç Artut is professor at the Education Faculty, Cukurova
University. Her current research focuses on mathematics education in teacher education,
teaching mathematics in elementary school, cooperative learning and problem solving. She
can be contacted at email: prhnrt@gmail.com.

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