Preschool Children’s Verbal Problem Solving Skills and the Types of Verbal Problems That Teachers Present to Children

Bu arastirmada, okul oncesi donem cocuklarinin sozel problem cozme becerilerinin problem turune ve problemdeki bilinmeyen turune gore incelenmesi ve okul oncesi ogretmenlerin cocuklara sunduklari toplama ve cikarmaya iliskin sozel problem turleri ile bu problem turlerinin uygulamasina iliskin goruslerinin belirlemesi amaclanmistir. Calismada karma yontem kullanilmistir. Arastirmanin orneklemini 5-6 yas grubundan 50 okul oncesi donem cocugu ile bu cocuklarin ogretmeni olan 6 okul oncesi ogretmeni olusturmaktadir. Arastirma da veriler sozel problem cozme testi ve yari yapilandirilmis gorusme formu kullanilarak toplanmistir. Veriler cocuklar ve ogretmenler ile bireysel gorusmeler yoluyla elde edilmistir. Arastirmanin sonucunda, cocuklarin sozel problem turlerinden ayirma, birlesim ve parca butun problem turunde, karsilastirma problem turune oranla daha basarili oldugu sonucuna ulasilmistir. Problemdeki bilinmeyen turune gore ise sonuc bilinmeyenli problemlerin degisim ve baslangic bilinmeyenli problemlere gore cocuklar acisindan cevaplamasi daha kolay oldugu sonucuna varilmistir. Ogretmenlerle yapilan gorusmeler sonucunda arastirmaya katilan ogretmenlerin, toplama ve cikarma islemlerinde daha cok sonuc bilinmeyenli problem turlerine yer verdikleri buna karsin cocuklara karsilastirma ve baslangic bilinmeyenli problemleri sunmadiklari ortaya cikmistir. Ogretmenlerin problem turlerinin hepsinden haberdar olmadiklari ve okul oncesi icin kullanilan yazili materyallerde butun problem turlerine yer verilmedigi icin bazi problemleri cocuklara sunmadiklari belirlenmistir.


Introduction
Problem solving takes place in many fields of education and it has an important place especially in mathematics education.Problem solving is one of the most important components of mathematics and it is also one of the most important parts of mathematics curricula (Van de Walle, 1994).Problem solving enables individuals to understand mathematics better and acquire the necessary skills for mathematics education (Yaman & Dede, 2005).In the process of problem solving, individuals are able to understand and use the information about mathematics, associate this information, develop new mathematical conceptions and find opportunities to explain different thinking styles (NCTM, 2000).In addition to this, problem solving skills cannot be developed in several weeks or months.The development of problem solving skills requires slow but frequent exposure and continuity (Van de Walle, 1994).Therefore, problem solving skills education should start at the pre-school period.
The presentation type of problems to the pre-school children has an important role in their solving the problems about addition and subtraction.The role of the language is really important in children's understanding the problems.The studies conducted show that the pre-school children are more successful when the problems are presented in the shape of verbal problems (Tarım & Deretarla Gül, 2003).
Children are able to solve the verbal problems about addition and subtraction in the informal mathematics education by developing their own strategies (Carpenter, Moser & Bebout, 1988).It is known that children start school with some skills about solving verbal problems.Children use the strategies of modelling or counting while solving these problems.On the other hand, children do not learn these strategies formally, they either learn them informally or discover by themselves (Carpenter,1985).
The verbal problems about addition and subtraction are classified in various ways.Some researchers (Carpenter & Moser, 1983;Van De Walle, 2001) classify these problems as joining, separation, partwhole and comparison.The problem solving skills of the children vary according to these types problems.In addition, the problem solving skills of the children also vary according to the type of unknown in the problems (Haylock & Cockburn, 2004;Nures & Brgant, 2008;Sarama & Clements, 2009).The teachers generally include joining and separation problems, namely addition and subtraction problems which can be solved in a single-step, in the pre-school period.Yet, it seems necessary to include sufficient number of problems from all types in the mathematics activities in order for the development of the children's verbal problem-solving skills (Peterson, Fennema & Carpenter, 1989).The conducted studies; however, show that the mathematics course books which were designed for elementary school children do not include some types of problems at all and the children are less successful at solving the problem types which are not presented in their course books (Olkun & Toluk, 2002).This might also be valid for the activity books designed for pre-school children.
When the studies based on verbal problems with addition and subtraction were considered, it was determined that the researchers focused mostly on the children's strategies of verbal problem solving and the children's learning behaviours on addition and subtraction (Bryant et al., 1999;Carpenter et al., 1993;Tarım & Deretarla Gül, 2003;Vilette, 2002;Wilkins et al., 2001).In a few studies, the effect of the method (cooperative learning) used in verbal problem solving was investigated (Artut;2009;Tarım, 2009).No studies; however, which were based on classroom implementations of the teachers -the most important actors in mathematics education-were found in the review of literature process.
Considering this fact, this research has two main purposes.The first one is to investigate the verbal problem solving skills of pre-school children about addition and subtraction.The second one is to determine the types of problems about addition and subtraction which are presented by pre-school teachers to children and the perspectives of pre-school teachers about the implementation of these types of problems.For this purpose, the following research questions were formulated; 1. What are the response rates of pre-school children about the addition and subtraction problems from joining, separation, comparison and part-whole type?
2. What are the response rates of pre-school children about the problems with initial unknown, change unknown and result unknown within the problem types of separation and joining?
3. What are the verbal problem types about addition and subtraction which pre-school teachers present to children?
4. What are the perspectives of pre-school teachers about the implementation of the addition and subtraction problems from separation, joining, comparison and part-whole types and from initial unknown, change unknown and result unknown problem types?
The results of this research are believed to support the findings in the related literature, the teachers as implementers and the experts who design the curricula.

Research Design
The mixed method was used in this research.In the mixed method, the researcher combines qualitative and quantitative methods, approaches and concepts.Creswell (2014) states that the mixed method studies contain the collection and analyze of qualitative and quantitative data in a single study or multiple studies.The reason of preference of qualitative methods in addition to quantitative methods is to provide contribution to obtain profound data about the topic and to make the topic be understood more comprehensively (Yıldırım & Şimşek, 2008).Descriptive research design was used in the quantitative dimension of this research and interview technique was used in the qualitative dimension of this research.

Participants
The population of this research is comprised of 5-6 aged children who attend pre-schools in Adana and the pre-school teachers who work in these schools.In order ro represent the population, three independent pre-schools affiliated with Adana Provincial Directorate of National Education were selected according to the suitable sampling method.50 children attending these schools were selected according to the random sampling method and the sample of this research was constituted.In addition, the teachers of the children who were taken into the sample of this research were also taken into the sample of this research.For this purpose, 6 teachers were included into this research.
Four of the teachers who were included into the research had almost 10 years of teaching experience (between 3 and 8 years) and two of them had almost 20 years of teaching experience (14 and 19 years).Six of the teachers had bachelor's degrees.Also, four of the teachers graduated from the Department of Pre-school Education and two of them graduated from the Department of Primary School Education.

Instrument
The data collection tools in this research were a "verbal problem test", developed by the researchers so as to evaluate the children's problem solving skills and to define the teachers' implementations about verbal problems and a semi-structured "interview form" for determining the perspectives of the teachers about the implementation of verbal problems.
Verbal Problem Test consists of 11 questions three of which were joining type (initial, change and result unknown), three of which were separation type (initial, change and result unknown), three of which were comparison type (difference unknown, larger unknown and smaller unknown) and two of which were part-whole type (whole unknown and part unknown).The verbal problem test was developed by the researchers based on the related literature.The test was broached to the experts primarily in order to provide its validity.Necessary revisions were made upon the suggestions of five experts who were teaching in the Departments of Pre-School Education and Primary School Education.Next, the test was presented to 10 children who were attending a pre-school and it was tested if the questions were understandable or not.Through this pilot implementation, it was aimed to eliminate the possible misunderstandings in the test and its implementation.As a result, necessary changes were made and the test was finalized.The internal consistency (KR20) was calculated so as to define the validity after it was implemented on the children in the sample.The results of the analysis showed that the internal consistency score of the test was .83,revealing that the test was reliable.
In order to collect data, the test was implemented on the children in the sample of the research and was presented to their teachers.The test was implemented to the children through concrete materials (such as legos, beads).
Semi-structured interview form consists of eight questions and it was formed by the researchers.In the form, there are open-ended questions about the verbal problem types of addition and subtraction which the teachers presented to children and about the reason why they preferred those types of problems.

Procedure
At the stage of data collection, the researchers went to the schools which were determined in advance and interviewed with the school directors and the teachers to explain the purpose of the research.Next, a letter explaining the purpose of the research and informed consent forms were sent to the parents so as to be able to do interviews with the children.The children whose parents filled in the informed consent forms and certified the participation of their children in the study were included into the research.The data were obtained through individual interviews with the children in a quiet room in the school.Before the test was given, the researchers had a short talk with the children in order to make them feel relaxed and they were asked if they wanted to answer some questions which were going to be asked or not.
The test was given to each child in the same order individually who accepted to answer the questions.The test took about 15 minutes for each child.The children were allowed to use concrete materials (legos) during the presentation of the verbal problems.After the interviews with the children were completed, they were thanked and taken back to their classroom.After the implementation of the test with the children in one class ended, the teacher of that classroom was interviewed individually.These interviews were made in a suitable place of the school rather than the classroom.The same verbal problem test which was given to the children was presented to the teachers and they were asked to tick the problem types which they used most in mathematics activities.Then, the questions in the semi-structured interview form were asked to the teachers in the same order.The interviews with the teachers took about 10 minutes.

Data Analysis
In the quantitative dimension of the research, the answers which were given by the students were evaluated as "successful" if they were correct and as "unsuccessful" if they were incorrect.For each correct answer, "one point" was given and the points taken from correct answers were calculated.They were presented as frequencies.
In the qualitative dimension of the research, the data obtained from the teachers were analyzed through content analysis.The process of content analysis started with the evaluation of the data transcribed.The aspects which became prominent or which were evaluated as important were defined and the codes were obtained following these themes.
In the research, the children were asked 11 types of questions in total from the types of joining, separation, comparison and part-whole about addition and subtraction.The answers of the children to these questions are presented in Table 1.

Results
The findings obtained in this research from the children and the teachers were presented in this chapter under two parts.  1 was considered, it was seen that the verbal problem type which the students found the most difficult was the "comparison" type problems.While 43 children could not answer any of the comparison problems, only 7 children could answer these questions correctly.When the number of the correct answers of the children who could answer these questions was investigated, it was found that one child gave correct answer to one question, two children gave correct answers to two questions and three children gave correct answers to three questions.

Verbal Problem Solving Skills of Pre-school Children
The problem types which were answered correctly the most were separation, joining and part-whole problems.While only one child could not answer any of the separation problems, four children could not answer the joining problems.The children were quite successful at part-whole problems; only two children could not answer this type of problems.The children did not generally have difficulty in answering part-whole problems.When the answers of 14 children who gave correct answer to only one of the part-whole problems were analyzed, it was revealed that 12 of them could answer whole unknown problems correctly and 2 of them could answer part unknown problems correctly.When the answers of the children to the initial unknown, change unknown and ending unknown problems from among separation and joining type problems were analyzed (Table 2), it was found that 34 of 50 children who participated in the study could not solve any of the initial unknown problems.Only six children answered initial unknown problems from both joining and separation type problems.3 of 10 children who could answer only one question gave correct answer to joining type of problems and 7 of them gave correct answer to separation type problems.
In the research, two questions with change unknown were asked to the children and it was observed that 18 of 50 children could not solve any of the problems with change unknown problems in the separation and joining type problems.11 children could answer change unknown problems in both separation and joining types.The rest 21 children gave correct answer to one question in only separation type problems.
When Table 2 was taken into account, it was found that only one of 50 children who participated in the study could not solve any of ending unknown problems while the other children could solve all of the ending unknown questions.45 of the children could answer two ending unknown problems correctly.One of the rest four children gave correct answers to the problems in joining type and three of them gave correct answers to the problems in separation type.

Verbal Problem Types that the Teachers Present to the Children
In the study, the teachers were asked about the verbal problem types that they presented to the children about addition and subtraction and their opinions about the implementation of these types of problems.The findings obtained in the research are presented in Table 3.

Table 3. The Teacher Distributions Regarding the Verbal Problem Types about Addition and Subtraction.
The teachers' answers n The number of the teachers who used only result unknown problems 3 The number of the teachers who used result and change unknown problems 1 The number of the teachers who used all types of problems mainly result unknown problems 2 In the research, the teachers were asked through "the verbal problem test" to mark the verbal problem types that they presented to the children in mathematics activities the most.When the findings were analyzed, it was determined that the teachers present result unknown problem type to the children the most.Three of six teachers who participated in the research stated that they used only result unknown (joining and separation) problem type among three problem types.The number of teachers who expressed that they presented both result and change unknown problem types to the children was one and the number of teachers who expressed that they presented all problem types but result unknown problem type more frequently to the children was two.In other words, all teachers who participated in this research used result unknown problems in mathematics activities the most.The teachers explained their opinions concerning this issue as follows: For example, one of the teachers explained his/her opinion about this issue as "I do not ask questions with initial and change unknown.I ask ending unknown questions" (T6).
Another teacher expressed his/her opinion as "I generally ask ending unknown questions, I do not ask initial and change unknown problems" (T3).
One of the teachers stated his/her opinion as "I use result unknown problems the total of which do not exceed 10.I have never asked and tried change and initial unknown problems" (T5).
Another teacher remarked his/her opinion as "I use ending and change unknown problem types more than the other types...I ask ending unknown problems more frequently" (T4).
When the opinions of the teachers who presented only result unknown problems to the children were analyzed, it was seen that the reason of their preferences about using only ending unknown problem types was prejudgements assuming that the children can not solve other problem types.The findings obtained from the interviews with the teachers showed that the teachers avoid using comparison type problems.Some teachers expressed that they consider comparison type problems unsuitable for the children without trying them.For example, one of the teachers explained his/her opinion about this issue as "Comparison type problems can be asked and the children can answer these problems.I have not asked larger unknown comparison questions but I do not think the children can give answers to these questions.It is a difficult problem type" (T6).
Part-whole problem type was not considered as a separate problem type by the teachers.It was considered as a similar problem type with joining and separation problem types.For example, one of the teachers verbalised his/her opinion about this topic as " I ask part-whole problem types in addition and subtraction (ending unknown)" (T6).
Besides, it was determined that the teachers who use all problem types use "initial unknown" and "comparison" type problems less than the other problem types.The teachers explained their reasons for not using some problem types as 'lack of resources containing various problem types (T3 and T6)' and 'not having faced this type of problem in their educational years (T3)'.

Discussion
There were two main purposes of this study.The first was to investigate the verbal problem solving skills of pre-school children.The second was to determine the types of problems about addition and subtraction which are presented by pre-school teachers to children and the perspectives of pre-school teachers about the implementation of these types of problems.The study contributes significantly into the related literature and provides guidance for the prospective researchers.
In the research, firstly, it was aimed to answer at what kind of verbal problem types the children are good.The results revealed that the children are more successful at joining, separation and part-whole problems than at comparison problems.It was also found that comparison problem types are the most difficult ones to answer for children.When the results were examined in a detailed way, it was determined that the children can solve separation problems more easily than joining problems.Furthermore, they can solve the problems with whole unknown more easily than the problems with part unknown.Similar findings are often reported in the related literature (Carpenter, Carey & Kouba, 1990;Cross, Woods & Schweingruber, 2009;Haylock & Cockburn, 2004;Nures & Brgant, 2008;Sarama & Clements, 2009).
There might be several reasons for the children's being less successful at separation, joining and part-whole problems than comparison type problems.As stated in the literature, it can be first because the teachers' frequent presentation of separation and joining problem types to the children (Nures & Brgant, 2008;Olkun & Toluk, 2002;Sperry Smith, 2001;Van De Walle, 2001).Secondly, it may result from not including some problem types in the written materials presented to children, as revealed in a study by Olkun and Toluk (2002) which was on the investigation of verbal problem types in elementary school mathematics coursebooks.Namely, the children never come across with these problem types.This can be the reason of the children's low achievement in some problem types that are not included in coursebooks.The third and the most important reason of the children's failure at solving comparison type problems can be developmental characteristics of the children.The children in this age group solve problems by modelling mostly through objects.Joining and separation problems are easier to model since they are generally in the way of combining or separating two independent clusters from each other.Therefore, it is believed that the children solve these types of problems more easily (Cross et al., 2009;Haylock & Cockburn, 2004;Nures & Brgant, 2008).
In this study, the children's problem solving skills about initial, change and ending unknown problems from among joining and separation type problems were also investigated.The results of the research showed that the unknown problem type in which the children have the most difficulty were the problems with initial unknown and change unknown, respectively.It was also found that the problem type which the children can solve the most easily were the problems with ending unknown problems.These findings are in line with the studies conducted on both elementary school students and pre-school students and the related literature (Carpenter et al., 1990;Cross et al., 2009;Haylock & Cockburn, 2004;Nures & Brgant, 2008;Sarama & Clements, 2009).In the study by Pilten (2010), which aimed at the evaluation of operational techniques that were used in solving problems with different semantic structures, it was found that the children were more successful at solving result unknown problem types than at change and initial unknown problem types.Olkun and Toluk (2002) obtained similar results in their study which was conducted with 2 nd , 3 rd and 4 th grade children in elementary school.
There might be several reasons why the children are unsuccessful at solving the problems with initial and change unknown which are presented to them.The first of them is the teachers' presentation of mostly problems with ending unknown during the mathematics activities as can be understood by the data obtained from the interviews with the teachers.Another reason, as stated by the teachers, might be the frequent presentation of the questions including problems with result unknown in the written materials produced for the children.Another important reason, as expressed by Piaget, might be the children's at this age group are cognitively unable to acquire everting skills.Since addition is the reverse of subtraction and subtraction is the reverse of addition, the children who haven't acquired this skill yet have limited ability in solving this type of problems (Aktaş Arnas, 2013).In the second main question of the research, the teachers' perspectives about the verbal problem types about addition and subtraction that they present to the children and the implementation of these types of problems were investigated.The results show that all of the teachers give place to the problems with result unknown the most in mathematics activities.The most important reason of the teacher's not using the other types of problems is their pre-judgements which assume that the children will not be able to solve these types of problems.
Similar results were observed when the related literature was reviewed.Limited number of studies which were conducted (Nures & Brgant, 2008;Sperry Smith, 2001;Van De Walle, 2001) show that preschool teachers do not present all types of problems adequately.The studies that were carried out with teachers reveal that the attitudes and beliefs the teachers have towards mathematics education affect the variety and quality of in-class implementations directly (Klibanoff & Levine 2006, Thompson, 1984;Ernest, 1989).In fact, the conducted studies show that there is a positive relationship between the mathematical knowledge of teachers and children's academic success (Hill, Rowan & Ball, 2005).Moreover, the results of this study revealed that most of the teachers are not informed about all types of verbal problems and they do not present some types of problems to the children as they are not included in the written materials which are used for pre-schools.It was confirmed by the studies in various countries in the related literature that mathematics course books do not include some types of problems or they are not equally presented.For example, in the studies of Parmjit and Teoh (2010) and Parmjit (2006) in Malaysia, Despina and Herikleia (2014) in Greece, Olkun and Toluk (2002) in Turkey, they found that mathematics coursebooks in the elementary education do not include some types of problems or they are not equally presented.These studies, however, were generally carried out in elementary education level and no data about pre-school education was observed.

Conclusion & Implementation
This study is believed to make significant contributions to the literature.The most important result of the study is the variation of the situation of the children's solving verbal problems according to the type of the problem and the unknown in the problem.It was concluded in this research that the children are more successful at separation, joining and part-whole types of problems from among verbal problem types than comparison type of problems.It was also revealed that the children solve the problems with ending unknown more easily than the problems with change and initial unknown in terms of the type of unknown in the problem.
The interviews with the teachers indicated that they do not prefer presenting comparison type problems and problems with initial unknown problems to children in mathematics activities.It was discovered that the majority of the teachers were not informed about all types of problems and they do not present some types of problems as those problems are not included in the written materials used for pre-school education.These results showed that there is a linear relationship between the types of problems that some teachers use in their programs and the children's success at solving those types of problems.
On the basis of these findings, it can be suggested that the teachers in the field should be informed about the existence of various types of problem and their instruction while organizing in-service training programs.Moreover, the writers of written materials for pre-school education should lay emphasis on this issue sensitively.Furthermore, it is believed that the awareness of the teachers about these types of problems might increase by adding these types of problems to the acquisitions while revising the preschool education curriculum.
Although this study makes significant contributions to the literature, it has also some limitations.Primarily, the results are not enough to make generalizations as the sample of this study is small in size.However, it presents important clues about both fields of mathematics education and pre-school education.In order to be able to make the results generalize, it can be suggested to the researchers that the prospective studies should be conducted with larger samples.In addition, this study has a descriptive design.Longitudinal and experimental studies are needed to reveal the lack of success of the children more clearly.For this purpose, it can be suggested that prospective studies should be designed in experimental research design and they should be longitudinal.

Table 1 .
The Frequencies and Percentage Distributions of the Children's Answers to the Verbal Problem Types about Addition and Subtraction.

Table 2 .
The Frequency and Percentage Distribution about the Answers by the Children on Initial Unknown, Change Unknown and Ending Unknown Problems among Separation and Joining Type Problems.