The Analysis of Pre-service Teachers’ Beliefs about Mathematical Problem Solving

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Procedia - Social and Behavioral Sciences 174 ( 2015 ) 2687 – 2692

Available online at www.sciencedirect.com

ScienceDirect

Peer-review under responsibility of the Sakarya University doi: 10.1016/j.sbspro.2015.01.953

INTE 2014

The

analysis of pre-service teachers’ beliefs about mathematical

problem solving

Gunes Yavuz, Hatice Nur Erbay

*

Istanbul University, Hasan Ali Yucel Education Faculty, Istanbul, Turkey

Abstract

There are many extracurricular and classroom factors that should be considered in developing problem solving skills of students in teaching Mathematics. Students who acquired the ability of problem solving will not only be successful in their lessons but also will have skills to overcome problems that they will experience in their real lives. Problem solving includes the combination and coordination of various skills, beliefs, attitudes, intuitions, knowledge and previous acquisitions. For this reason, it has a key role in teaching mathematics. The purpose of this study is to analyze pre-service teachers’ beliefs about mathematical problem solving in terms of various variables. For this purpose, the data was collected from 310 third-year students who are studying in teaching mathematics, classroom teaching and teaching science departments from two public universities in Istanbul. Belief Scale about Mathematical Problem Solving was used as a data collection instrument (Haciomeroglu, 2011). The data were analyzed by using statistical software. The correlation of some variable to pre-service teachers’ belief about mathematical problem solving was analyzed.

Peer-review under responsibility of the Sakarya University. Keywords: problem solving, pre-service teachers, belief

1.Introduction

The social structure which is getting increasingly complex and technological developments, political, social and economical crises make an individual to encounter with constantly increasing problematic situations. Therefore, problem solving is an important issue which becomes the centre of attention in psychology for many years. Many

* Corresponding author. Tel: +90 553 463 3053

E-mail address: [email protected]

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concepts have been introduced about problem solving process in many studies. These concepts include a variety of features from traditional Gestalt approaches among different learning methods to recent computer simulations and mathematical models.

It was seen that while researchers such as Gagne and Skinner (1964; 1974) tended to analyze the previous lives of individuals as the most important variable in problem solving process, the other researchers such as Kohler and Maier (1925, 1970) advocated individuals’ perception regarding the experienced situation as the most important factor in solving a problem.

A problem is defined as “a situation, quantitative or otherwise, that confronts an individual or group of individuals, that requires resolution, and for which the individual sees no apparent or obvious means or path to obtaining a solution” (Polya, 1962). The problem from a mathematical perspective is defined as “a situation where something is to be found or shown and the way to do this is not immediately obvious to the solver” (p.218, Grouws, 1996, cited in. Kayan & Cakıroglu, 2008). And mathematical problem solving is expressed as a situation which occurs from the experiences of the students (Schoenfeld, 1985). Problem for a mathematics teacher means an outstanding question which students have the necessary pre-information but which they do not know the ways and steps which will lead them to solution beforehand (Schoenfeld, 1989). From this perspective, problem solving means not only finding the solution of a mathematical problem but also confronting with new situations and finding flexible, useful and elegant solutions to these situations (Gail, 1996).

Gur and Korkmaz (2003) emphasize that posing and solving problems enable students both to gain experiences regarding how to express mathematical ideas in written or verbal and to discover mathematical situations. Students’ skills to use problem solving strategies regarding mathematical problems are helpful to them for making correct choices for the situations that they encountered in their daily lives (Altun, 2008).

In recent years, it is seen that problem solving is considerably emphasized in studies for improving teaching primary mathematics programs both in Turkey and worldwide.

Mathematical beliefs are defined as value judgements which are composed of the past experiences of an individual (Raymond, 1997). These beliefs have an important place in teaching and learning process as they affect the perceptions of individuals (Pajares, 1992; Thompson, 1992). For instance, the beliefs of a teacher is effective in his/her immediate decisions when he/she needs to make a decision in a difficult situation that he/she faces during teaching and learning process (Abrosse, Clement, Philipp & Chauvot, 2004). Kloosterman and Stage (1992) points out the fact that mathematical beliefs of an individual effect learning and problem solving. Concordantly, the studies which analyze the beliefs about mathematics shows that students perceive problem solving as finding the correct answer and think that learning mathematics requires memorizing (Picker & Berry, 2000; Raymond, 1997; Schoenfeld, 1989; Thompson, 1984; Toluk Ucar, Piskin, Akkas & Tasci, 2010).

Hart (2002) analyzed the effects of mathematical prospective teachers’ beliefs about teaching mathematics. The findings show that the beliefs of prospective teachers advance positively at the programs that they studied. Accordingly, Kayan and Cakiroglu (2008) analyzed the beliefs of prospective primary school mathematics teachers regarding mathematical problem solving. It was determined that prospective teachers had positive beliefs about problem solving. Besides, it was specified that prospective teachers had internalized traditional views such as the importance of routine calculation skills in teaching mathematics and the necessity of following the pre-specified steps in problem solving. The studies (Frykholm, 2003; Kayan & Cakiroglu, 2008; Lloyd & Wilson, 1998) show that the beliefs of prospective teachers about mathematical problem solving is an effective factor in organizing learning environments as a teacher and in the achievements of students. For this reason, determining the beliefs of prospective teachers will be informative about the practice of a future teacher (Kayan & Cakiroglu, 2008).

In this regard, the main purpose of this study is to analyze the beliefs of prospective teachers about problem solving. The beliefs are defined in terms of educational studies as basic assumptions, arguments and insights that an individual has in his/her mind and that he/she develops for the facts around him/her (Richardson, 1996). The beliefs of teachers; is an important factor affecting both the classroom environment that they created and the beliefs of students about mathematics (Ball, 1998; Grouws, 1996; Schoenfeld, 1992; Wilkins & Brand, 2004).

For this reason, analyzing the beliefs of prospective mathematics teachers and having an extensive understanding about them will enlighten teaching mathematics and the studies regarding improving students’ learning and will contribute to the training studies which will be organized for teachers. The research subject in this study was worth investigating for the reasons stated above.

The purpose of this study is to analyze the beliefs of prospective teachers who are studying in teaching mathematics, classroom teaching and teaching science departments about problem solving. In this sense, the

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answers of the following sub-problems were searched:

1. How are the beliefs of prospective teachers about problem solving?

2. Do the beliefs of prospective teachers about problem solving vary according to universities that they study? 3. Do the beliefs of prospective teachers about problem solving vary according to their departments? 4. Do the beliefs of prospective teachers about problem solving vary according to the types of high schools

that they graduated from?

5. Do the beliefs of prospective teachers about problem solving differ according to gender?

2.Method

The descriptive survey model among qualitative research methods was used in this study. The descriptive survey is suitable for describing a case which exist in the past or present without influencing it (Karasar, 2003). Since the purpose of this study was to present the beliefs of students who are studying in teaching mathematics, classroom teaching and teaching science about mathematical problem solving, descriptive survey model was used.

Study Group: The sample of the study is composed of third-year prospective teachers who are studying in

educational faculty, teaching mathematics, classroom teaching and teaching science and technology majors of two different universities in 2013-2014 academic years. The distribution of the sample group can be seen in Table 1.

Table 1: Study Group

Majors 1st University 2nd University Tot a l F M F M Teaching Mathematics 50 6 32 5 93

Teaching Science and Tech. 44 6 43 7 100

Classroom Teaching 39 13 49 16 117

Total 133 25 124 28 310

Data Collection Instruments: Belief Scale about Mathematical Problem Solving was developed by

Kloosterman and Stage (1992) and adopted to Turkish by Haciomeroglu (2011). This scale was developed for revealing the beliefs of students about mathematical problem solving. When the whole scale was analyzed, the Cronbach alpha internal consistence coefficient was calculated as 0.768. The scale was consisted of 24 items and it was in five point likert type.

3.Findings

In order to search for an answer for the first sub-problem, prospective teachers’ average scores regarding beliefs about problem solving was displayed in Table 2.

Table 2: Average Scores of Prospective Teachers’ Beliefs about Mathematical Problem Solving

N _X SS

310 81 5,83

The beliefs of prospective teachers about mathematical problem solving were collected through a likert type form which was scored between 1 and 5 and consisted of 24 items. The total score differed between 24 and 120. The findings show that the level of beliefs of prospective teachers about problem solving is high (

X

=81).

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The first item of the findings regarding the second sub-problem was displayed in Table 3.

Table 3: T-Test Results of Prospective Teachers’ Belief Scores about Mathematical Problem Solving according to their Universities

As it can be seen from Table 3, there was not a significant difference between the beliefs of prospective teachers about mathematical problem solving according to their universities (p>0.05). This finding showed that the beliefs of prospective teachers about mathematical problem solving were not affected by the university that they studied.

The average scores of the students were analyzed in order to understand whether or not the level of their beliefs about mathematical problem solving varied according to their majors and the findings obtained were given in Table 4.

Table 4: T-Test Results of Prospective Teachers’ Belief Levels about Mathematical Problem Solving According to their Majors

Majors N _X SS F P

Teaching Mathematics 93 83,26 4,89

7,79 0,00 Teaching Science and Tech. 100 80,19 5,63

Classroom Teaching 117 80,82 6,34

Total 310 81,35 5,83

As it is seen in Table 4, there was a statistically significant difference between average belief scores of prospective teachers about mathematical problem solving as the average scores of teaching mathematics students were higher than teaching science and classroom teaching students (p<0.05). The reason of this difference was because of the fact that the average belief scores of mathematics prospective teachers about mathematical problem solving were higher.

The other sub-problem was ‘Do the beliefs of prospective teachers about problem solving vary according to the types of high schools that they graduated from?’ The findings regarding this sub-problem are given in Table 5.

Table 5: F-Test Anova Results of Prospective Teachers regarding the Belief Levels about Mathematical Problem Solving according to the type

of the high school

Type of High School N _X SS F p

Anatolian High School 124 81,04 5,58

2,83 0,06 Teacher Training High School 81 82,65 5,83

Other high schools 105 80,72 6,03

Total 310 81,35 5,83

According to this finding, it can be said that the level of prospective teachers’ beliefs about mathematical problem solving did not differ depending on the type of high school that they graduated (p>0.05).

Whether or not the level of prospective teachers’ beliefs about mathematical problem solving differed according to their gender is given in Table 6.

University N _X SS T p

The First Unv. 158 81,00 5,91

1,09 0,27 The Second Unv. 152 81,72 5,75

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Table 6: T-test results of Prospective teachers’ Belief Levels about mathematical Problem solving according to their Gender

According to Table 6, there was not a statistically significant difference between the average belief levels of prospective teachers about mathematical problem solving by their gender. (p>0.05)

4.Conclusion, Discussion and Suggestions

The basic purpose of this research is to determine the beliefs of prospective teachers about mathematical problem solving in terms of various variables. Within the scope of this basic purpose, the following results have been obtained.

It can be said that the beliefs of Mathematics, Science and Classroom Teaching prospective teachers about mathematical problem solving is generally high. This finding of the study is supporting Kayan and Cakiroglu’s (2008) study in which they stated that primary mathematics prospective teachers had positive opinions about problem solving.

It was concluded that beliefs of prospective teachers about mathematical problem solving did not differ according to their universities. The reason of this situation may be the fact that both universities are located in the same region. Different universities may not have an effect since students’ environments do not differ in general terms. It is necessary to repeat this study with a broader sample to see whether or not different universities have an effect by selecting universities from different regions of Turkey.

When it was analyzed whether beliefs of prospective teachers about mathematical problem solving differed according to their majors or not, it was seen that the difference was in favour of students from teaching mathematics major. It can be said that the beliefs of mathematics prospective teachers about problem solving is higher than science and classroom teaching prospective teachers. The reason of this situation may be the fact that problem solving is occupationally vital for the mathematics prospective teachers. Besides, the university education of mathematics prospective teachers may have an effect on their problem solving skills.

The beliefs of prospective teachers about mathematical problem solving did not differ according to the type of high school that they were graduated from is another finding of the study. The reason of this may be the fact that educational faculties are mostly preferred by Anatolian High School or Teacher Training High school graduates. Since a few of the prospective teachers were graduated from science high schools, private high schools or vocational high schools, this number was not enough for creating a difference. In other words, it cannot be talked about any kind of difference since a homogenous sample could not be created according to high schools. By using purposeful sampling, it can be possible to understand whether this situation will create a difference or not.

The beliefs of prospective teachers about mathematical problem solving did not differ according to gender is the final finding of the study. Kayan (2007) concluded in his study which was carried out in five universities that the beliefs of primary mathematics prospective teachers about problem solving did not differ according to gender but Soyturk (2011) concluded in his study about classroom teaching prospective teachers that the beliefs of prospective teachers about problem solving differed in favour of female students. When the literature was analyzed, it was stated that whether or not the beliefs of female and male students about problem solving differed according to gender might change according to sample group (Arli, Altunay & Yalcinkaya, 2011).

The following suggestions can be offered in accordance with the findings.

1. The beliefs of prospective teachers about problem solving can be searched with different and broader samples. The sample can be decided by considering different geographical regions.

2. Comprehensive studies can be carried out in which the beliefs of prospective teachers about mathematical solving can be evaluated in terms of gender.

Gender N _X SS T p

Female 257 81,55 5,6

1,34 0,18 Male 53 80,37 6,45

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3. More comprehensive studies can be carried out by considering other subject teachers.

4. Studies based on qualitative data about the beliefs of prospective teachers about mathematical problem solving can be carried out. In this regard, prospective teachers can be evaluated by considering separately each sub-factors of the ‘Belief Scale about Mathematical Problem Solving’ that was used in this study.

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