# TEACHERS’ CONCEPTIONS AND BELIEFS ABOUT MATHEMATICAL PROBLEM SOLVING IN RELATION TO THEIR CLASSROOM INSTRUCTION

TEACHERS’ CONCEPTIONS AND BELIEFS ABOUT MATHEMATICAL PROBLEM SOLVING IN RELATION TO THEIR CLASSROOM INSTRUCTION

ABSTRACT
The purpose of this study was to determine secondary school mathematics teachers’ conceptions and beliefs about mathematical problem solving in relation to classroom instruction. Seven research questions and eight hypotheses guided this study. The study was conducted in Anambra and Enugu States. The target population was all secondary school mathematics teachers in the two states. The sample was 327 secondary school mathematics teachers. They were drawn from 61 secondary schools, through a multi-stage sampling procedure that consisted of stratified, simple random, and purposive sampling techniques. Two instruments – Problem Solving Self Administered Questionnaire (PSSAQ) and Classroom Instruction Observational Checklist (CIOC) – were developed and used for data collection. Descriptive analysis involving means, standard deviations, and percentages were used to answer research questions. The null hypotheses were tested at 0.05 alpha-level using the following statistical tests; multiple regression, analysis of variance (ANOVA), t-tests, and chi square tests.
The results showed that mathematics teachers have varied conceptions of problem solving with the most dominant ones being problem solving as practice and drill exercises, working with word or application problems, an instructional approach, and a process of finding a solution to an unfamiliar situation. The following beliefs about problem solving were identified: teachers would teach problem solving strategies if they are incorporated in mathematics textbooks; problem solving is an everyday activity; problem solving is the main reason of teaching mathematics in secondary schools; teaching students how to use problem solving skills and strategies is a good way to teach mathematics, teaching problem solving in schools can improve students’ performance in WAEC mathematics exams. The results also showed that many teachers do not present non-routine problems during problem solving but rather employ a number of problem solving strategies such as breaking a complex problem into a simpler form and solving same, restating a problem in students’ own words, using pattern recognition and inductive reasoning, and drawing a diagram. The results further showed that mathematics classrooms are often teacher dominated with the teacher talking about 85% of the class time. Additionally, there is a significant relationship between teachers’ conceptions and beliefs with what they teach. However, teachers’ conceptions and beliefs were not significantly related to how they teach. The results also showed that teachers who have less than three years of experience and those with more than 10 years experience have different conceptions than those who have between seven and nine years experience. This study has implications for classroom teachers, curriculum developers, textbook writers and teacher preparation institutions.

CHAPTER ONE

INTRODUCTION
Background to the study
Knowledge of mathematics is crucial for the social, economic, and technological development of any nation as well as for the day-to-day activities of individuals and groups. In connecting mathematics to Nigerian technological development, Kalejaiye (1979, 1989) noted, the level of technological development that a nation can reach depends on the quality of its mathematics education. In the same manner, Ukeje (1997) acknowledged that mathematics is central to the modern culture of science and technology. He described the relationship between mathematics and a modern society in terms of linear steps, arguing that without mathematics there will be no science; without science there will be no modern technology; and without modern technology there will be no modern society. Other mathematics educators had reiterated the need for mathematics education in Nigeria to be strengthened, to make it relevant to the country’s economic and technological development. Aguele and Usman (2007) argued that what Nigeria needs to be able to meet the challenges of the 21st century is a strong foundation in mathematics that can effectively put science and technology in the fore-front of its nation building. Probably for this reason, one of the objectives of secondary education under the National Policy of Education (2004) is to equip students to live effectively in a modern age of science and technology. In addition to being the foundation of science and technology, mathematics plays other important roles. It is highly related to other disciplines; it sharpens critical thinking; and nurtures an individual’s reasoning and problem solving abilities (Agwagah, 2005).
In spite of all the benefits and relevance of mathematics; many students still have negative attitudes and lack interest in the subject (Esan, 1999; Amazigo, 2000). Students also have great anxiety towards mathematics (Omosigho, 2002) and consequently perform poorly in mathematics external examinations (Adeleke, 2007). Poor performance in mathematics has been of great concern to teachers, parents, mathematics educators, and researchers in general. They are particularly worried about students’ poor performance in the Senior School Certificate Examination (SSCE) conducted by the West African Examination Council (WAEC) and the National Examination Council (NECO). The poor performance according to Agwagah and Usman (2004) and Adeleke (2007) has now become a yearly affliction. This was further reiterated by This Day’s (April 9, 2009) editorial report which stated that there has been a consistent poor performance in mathematics at these national examinations for about 10 years. This claim was confirmed by available statistics of students’ SSCE results from WAEC 1995 to 2008 which showed that on the average, less than 43% of students who sit for SSCE examinations pass at credit level or above. The situation seems to be getting worse in spite of the fact that the most recent education policies and programs are directed towards science and mathematics.
Many researchers and individuals have analyzed reasons for this poor performance and proffered solutions. For instance, about three decades ago, some scholars such as Aina (1986), Olorundare (1989), and Ale (1989) specified among other things, that the reason for persistent poor performance in mathematics was because problem-solving skills were not given adequate attention in our secondary schools. Over these years, problem solving has remained the focal point of mathematics instruction in both developed and developing nations. In Nigeria, Odili (2006 p. 109) acknowledged that “problem solving found its way back into the 1989 mathematics curriculum as the central focus of mathematics education in Nigeria” but he concurred with earlier reports that even so, “little change occurred in the teaching of problem solving in the mathematics classroom.” Odili and other mathematics educators in Nigeria recommended that problem solving should be made an important component of mathematics classroom activities. In the United States of America, the National Council of Teachers of Mathematics (2010) stated that problem solving should not be an isolated part of the curriculum, but a rich part of what it means to do and understand mathematics.
There are generally two schools of beliefs about mathematical problem solving. The first school is composed of those who view it as the ability to use previous knowledge to solve unfamiliar problems. This is the sense in which Odili (2006, p. 115) described it as a “process that requires the learner to sift through previously acquired knowledge, and select an appropriate plan in solving the problem”. This process may entail applying heuristics, which are general solution strategies that may not necessarily lead to correct answers. Using heuristics requires the problem solver to (a) develop a clear understanding of the problem he or she encounters (b) translate the problem from everyday language into a precise mathematical question; (c) choose and use appropriate methods to answer the question; and (d) interpret and evaluate the solution in terms of the original problem bearing in mind that not all problems can be solved mathematically.
The other group of scholars such as Posamentier and Krulik (1998) and Wilson, Fernandez, and Hadaway (1993) described problem solving as an instructional approach that incorporates exploration, discovery, and creation of algorithm in mathematics teaching and learning. According to this perspective, mathematics should be taught through problem solving which demands that students learn through reflective thinking and try to discover the mathematical content themselves by working and interacting with each other. It is in this context that Taplin (2009) outlined the six specific characteristics of teaching mathematics through problem solving as: (1) interactions between students/students and teachers/students; (2) mathematical dialogue and consensus between students; (3) teachers providing just enough information to establish background/intent of the problem, students clarifying, interpreting, and attempting to construct one or more solution processes; (4) teachers accepting right or wrong answers in a non-evaluative way; (5) teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems; and (6) teachers knowing when it is appropriate to intervene, and when to step back and let the students make their own way.
For the purposes of this study, problem solving is regarded as: 1) the goal of teaching mathematics, 2) the process of using previously acquired knowledge to solve an unfamiliar problem, and 3) an instructional approach where teachers provide an environment that allows students to try new things, explore, take risk, interact and argue with each other while working through challenging problems.
In support of research results (Agwagah, 1993; Alio, 1997; Adigwe, 1998; Adeleke, 2007; Kolawole & Ilugbusi, 2007) that seemed to indicate that teaching mathematics through problem solving leads to better understanding and improved performance, the WAEC Chief Examiner (1999) affirmed in his report that students should be equipped with problem-solving skills through proper practice and also be encouraged to develop a problem-solving approach to mathematics. His report indicated that effective problem solving skills would boost students’ confidence and help them prepare for national examinations. Also WAEC Chief Examiner’s report of 2005 and 2008 attributed students’ poor performance in mathematics to students’ lack of skills to tackle mathematical problems. These skills which they lack, could be problem solving skills. As Adeleke (2007) pointed out, one way to improve students’ performance in mathematics is to teach mathematical problem solving skills early enough in the education of the students. Also Setek and Gallo (2005) argued that mathematical anxiety and lack of problem solving skills are the two serious issues that must be addressed in any attempt to help students to develop and sustain self-confidence, critical thinking, and better performance in mathematics. It follows therefore that making problem solving the locus of Mathematics instruction by teaching necessary skills is a good strategy for improving performance in mathematics.
A number of teaching approaches to problem solving have been recommended by mathematics educators since problem solving became topical in mathematics education. Early studies on teaching problem solving in schools focused on Polya’s (1957, 1962) four-step framework that involves 1) understanding of the problem, 2) devising a plan, 3) carrying out the plan, and 4) checking the reasonableness of the solution. Some studies have documented positive effects of teaching mathematics by following Polya’s model (Agwagah, 1993; Alio, 1997; Adigwe, 1998; Kolawole & Ilugbusi, 2007). In some cases, the studies emphasized teaching of reading and comprehension as a means of understanding the problem (Agwagah, 1993), while others focused on applying Polya’s four general stages (Alio, 1997), or teaching specific strategies (Adigwe, 1998; Kolawole & Ilugbusi, 2007). The effectiveness of these approaches would depend, to some extent, on the classroom teachers’ conceptions, beliefs, and classroom instructions as well as their acceptance of research recommendations.
In Nigeria, considerable efforts have been made to make problem solving the center of mathematics teaching and learning (Aina, 1986; Ale, 1989; Olorundare, 1989; Agwagah, 1993; Alio, 1997; Adigwe, 1998; Adeleke, 2007; Kolawole & Ilugbusi, 2007). Yet, little is known about what teachers know about problem solving and the extent they believe in problem solving strategy and therefore are themselves committed to teaching mathematics through problem solving. The aforementioned studies on problem solving have focused primarily on identifying effective pedagogies but did not investigate the extent to which those at the center of the implementation are conversant with the approach and believe in it. Therefore, one research area that appears to be neglected in all efforts to improve students’ performance in mathematics through problem solving is the role of classroom teachers, more specifically, their conceptions and beliefs of problem solving. The classroom teacher is an important factor for any meaningful reform and classroom improvement since he or she is the pivot on which classroom activities rest, as he or she has direct contact with students (the learners). This implies that, as Griffiths and Howson (1974) noted, the success or failure of any innovation in education such as problem solving in mathematics ultimately hinges on the receptiveness and flexibility of the classroom teacher. The receptiveness and flexibility of the classroom teacher in relation to problem solving in mathematics will depend on his/her conceptions and beliefs of problem solving.
Unfortunately research on classroom teachers’ conceptions and beliefs about innovations such as problem solving is lacking in our mathematics education research domain even though this type of research has been going on in other countries for some time. In addressing the need to extend research activities to study teachers’ behavior, beliefs and conceptions, Thompson (1984, p. 106) wrote:
If teachers’ characteristic patterns of behavior are indeed a function of their views, beliefs, and preferences about the subject matter and its teaching, then any attempt to improve the quality of mathematics teaching must begin with an understanding of the conceptions held by the teachers and how these are related to their instructional practice. Failure to recognize the role that the teachers’ conceptions might play in shaping their behavior is likely to result in misguided efforts to improve the quality of mathematics instruction in the schools.

Teachers’ conception of mathematical problem solving in the context of this study is the general knowledge, the understanding of the specific meaning, or mental structure (Thompson, 1992) teachers have about problem solving that can shape their classroom instructions in terms of what they teach, and how they teach. Teachers’ beliefs on the other hand are teachers’ feelings, trust, confidence and attitudes about problem solving that equally play an important role in classroom teaching and learning of mathematics. This implies that there is a direct relationship between teachers’ beliefs and conceptions and their instructional strategies (Grouws & Goods, 1989; Pajares, 1992). Suffice it to posit that understanding teachers’ beliefs and conceptions as well as their classroom practices may be an important step in improving classroom instructions. According to Richardson (1990) and Schmidt and Kennedy (1990), it indicates that teachers’ conceptions and beliefs are predisposing factors that can significantly impact their teaching effectiveness. Hence there is need for this study.
Two aspects of teachers’ conceptions – teachers’ general conceptions and teachers’ context-specific conceptions – are of interest in mathematics education research. Teachers’ general conceptions are the values and beliefs that can impact their instruction. These include such things as teachers’ beliefs about teaching and learning, their beliefs about the subject they are teaching, their beliefs about curriculum and what to teach and their beliefs about their pedagogical approach (Calderhead, 1996). Context-specific conceptions refer to knowledge about specific mathematical content and knowledge about how to teach specific topics to particular students (Henderson, 2002). This study integrated the two areas in determining secondary school teachers’ overall conceptions and beliefs about mathematical problem solving to determine how these could be related to their classroom instruction and ultimately affect students’ mathematical achievement.
Teachers’ conceptions and beliefs influence teachers’ decisions about what to teach and how to teach; and these conceptions and beliefs could be influenced by factors such as teachers’ teaching experience, teachers’ academic qualifications, and teachers’ gender. Teachers’ teaching experience as applied in this study is the length or period of time teachers have engaged in the teaching and learning of mathematics. Even though teachers are said to be born and not made, which implies that the personal ability and traits of teachers are more important for teaching effectiveness than pedagogical training and experience; yet, studies have shown that teachers are significantly more effective when they have at least two years experience and if they entered the profession with adequate preparation (Berry, Daughtrey, & Wieder, 2009). In general, teacher educators believe that teachers develop their beliefs, values and teaching skills during their teaching experience (Cooney, 1985; Marks, 1987; Schmidt & Kennedy, 1990; Erickson, 1993). Although conceptions and beliefs, therefore, could be acquired through the study of mathematics, they are mostly developed during teaching practice or at the place of work. Cooney (1985) and Marks (1987) found differences between experienced teachers’ classroom behavior and the behavior of inexperienced teachers when dealing with problem solving. The same argument is made about teachers’ beliefs. Schmidt & Kennedy, (1990) and Erickson, (1993) argued that novice and experienced teachers hold different mathematical beliefs, which is an outcome of their personal experience. The paucity of studies in Nigeria relating teachers’ teaching experience to their conceptions and beliefs of problem solving necessitates the conduct of this study.
Mathematics is generally viewed as a male domain discipline (Frieze & Hanusa, 1984; Cole & Griffin, 1987) thus, it is widely accepted that male students perform better than female students in mathematics examinations. This notion extends to problem solving where Alio and Harbor-Peters (2000) found that male students performed better on tests of mathematical problem solving than female students. Other studies have come to a similar conclusion on the existence of gender differences in mathematical problem solving (Linn &
Petersen, 1985; Ben-Chaim, Lappen & Houang, 1988; Tartre, 1993; Tartre & Fennema, 1995; Royer, Tronsky, Chan, Jackson, & Marchant, 1999; Gallagher, DeLisi, Holst, McGillicuddy-DeLisi, Morely, & Cahalan, 2000). However, Adeleke (2007) argued that the gender difference does not exist when problem solving is taught using Conceptual and Procedural Strategies which means that boys and girls would perform equally well without any significant difference. However, studies on gender differences in mathematical problem solving usually target students. There seems to be no study yet that addresses teachers’ gender differences in any form. This study therefore, sought to determine whether or not there are differences between male and female teachers’ conceptions of and beliefs about mathematical problem solving.
It is pertinent to note that teachers may have correct conceptions and beliefs but fail to communicate effectively to students because of their pattern of interaction in the classroom.
Interaction pattern is the frequency of verbal exchanges teachers have with their students during an instructional process. Walberg (1986) analyzed key factors that result in effective teaching and showed that the key elements are: engaged academic learning time, use of positive reinforcement, cooperative learning activities, positive class atmosphere, higher-order questioning, feedback, and use of advance organizers. These elements fit together in actual classroom interaction and the structure encourages active participations and interactions between and among students, interactions between students and teachers, and opens up mathematics dialogue where students try to come to a consensus on ways and means of tackling mathematical problems. Classroom interaction pattern accounts for the flow of communication between teachers and students during teaching and learning. Interaction pattern in the context of this study is measured by how often teachers talk to students, students talk to teachers, students talk to students, teachers ask and/or answer questions, students ask and/or answer questions, teachers give directives, students suggest ideas, and teachers’ praises or encourages students’ actions.
In a typical problem solving classroom environment, the teacher is mainly a coach who provides guidance and encourages dialogue in the classroom through an appropriate and effective instructional pattern. An instructional pattern is a method for delivering instruction that is intended to help students achieve a learning objective (Burden & Byrd, 1999). Some instructional patterns such as “whole class lecturing” are more directed and teacher centered while others such as “group work and discussions” are more interactive. For the purposes of this study, instructional patterns deal with the organizational structure of the classroom and they include whole class, small group and cooperative, teachers working with students one-on-one, and individualized seat work. Studies have shown that active student participation in learning (Salman, 2009) as well as cooperative or group work where students can explain their thinking while solving problems (Alediosu, 1998; Esan, 1999) are important components of problem solving.
In addition to instructional pattern, the type of problems teachers present in class and problem solving strategies employed by teachers can lead to effective learning if students are properly engaged. According to Hiebert, Carpenter, Fennema, Fuson, Human, Murray, Olivier, and Wearne (1996), engaging students in a problem solving situation will entail providing a variety of problem situations for students to search for solutions by using reflective methods or what they called “problematizing the situation”. In the context of this study, the categories of problems presented during mathematics instruction are regarded as problem types and they are identified as exercises, routine problems, non-routine problems, application problems, word problems, and puzzles, problems to prove. Research shows that real world problems that require a higher level of analysis in the classroom enhance students’ problem solving ability (Lampert, 1990; Schoenfeld, 1992; Jitendra, DiPipi, & Perron-Jones, 2002). Other studies have specifically recommended non-routine problems in a problem solving instruction. This is because such problems pose challenges to students and demand the application of multiple steps and may permit a variety of solutions (London, 2004). Generally, students mimic the problem solving strategies their teachers employ when solving mathematical problems. Problem solving strategies are the specific plans and tactics teachers use when solving problems. These strategies are identified in this study as working backward, finding a pattern, systematic listing, trial and error, inductive reasoning, deductive reasoning, drawing a picture, solving similar or analogous problems, restating the problem, making a table, and breaking problems into simpler forms.
It is obvious from the foregoing background that students’ performance in mathematics cannot be improved by merely including problem solving in the secondary school mathematics curriculum or simply by recommending that teachers teach mathematics through problem solving. Rather, a significant positive change can be made by ensuring that students are taught problem solving strategies; that they are properly engaged in mathematics classrooms through active participation and cooperative learning; that they are challenged through the presentation of non-routine problems; and that the teachers guide them rather than prescribe what to do. These are the conditions that leverage students’ learning of mathematics successfully through problem solving. Thus, the locus for change resides in the classroom. If any meaningful change is going to take place, the culture of the classroom has to change significantly, and any classroom change must begin with the teachers (Hiebert et al., 1996).

Statement of the problem
It has been suggested from evidence that students’ underachievement in mathematics may be attributed to many factors and that lack of problem-solving skills is paramount. Problem solving has been widely acknowledged to be at the core of mathematics teaching and learning. It can enhance students’ understanding of mathematical content and ultimately improve students’ performance. Probably for this reason, mathematics educators recommended, two decades ago, that students should be taught problem solving strategies. However, evidence from literature as well as from WAEC and NECO results show that most students are yet to acquire the vital problem-solving skills required to be successful in mathematics and related disciplines. It is possible that researchers’ recommendations are not being implemented in the mathematics classrooms.
The non-implementation of these recommendations may be attributed to teachers’ conceptions and beliefs about problem solving. From the literature, it has been shown that some mathematics teachers who do not have belief in the efficacy of problem solving strategies do not spend time showing their students how to solve problems. Similarly, teachers’ wrong conceptions of problem solving tend to be apparent in what they teach and how they teach. Therefore if teaching problem solving should be a major component of secondary school mathematics instruction as recommended by researchers and mathematics educators, then it is necessary to establish what teachers know, what they do, and how they feel about mathematical problem solving as these may be influencing what they teach and how they teach mathematics.
Surprisingly, not many studies, if any, in Nigeria have attempted to investigate teachers’ conceptions and beliefs about mathematical problem solving. So what are the conceptions and beliefs of Nigerian secondary school mathematics teachers about problem solving? What are the relationships between such variables as teaching experience, gender, and academic qualifications and these teachers’ conceptions and beliefs about problem solving? These questions underlie the problem of this study

The purpose of the study

The main purpose of this study is to determine teachers’ conceptions and beliefs about mathematical problem solving and how these conceptions and beliefs are related to how they teach and what they teach. Specifically, the study intends to:
1. Determine secondary school teachers’ conceptions regarding mathematical problem solving.
2. Determine secondary school teachers’ beliefs about mathematical problem solving.
3. Determine the current situation on teaching mathematics through problem-solving in secondary schools.
4. Determine problem-solving strategies that mathematics teachers employ when
teaching mathematics or when solving problems for students.
5. Find out the types of problems teachers solve in class during mathematics instruction.
6. Determine whether gender, academic qualifications, and teaching experience have any relationship with teachers’ conceptions and beliefs about mathematical problem solving.
7. Determine the nature of instructional and interaction patterns in a typical secondary school mathematics classroom.

Significance of the study
When published, the result of this study will be beneficial to mathematics teachers, students, researchers, teacher educators, textbook writers and curriculum planners, wider society, business communities and all those who use computations and figures, and education policy makers.
The outcome of this study will help mathematics teachers recognize that wrong conceptions about, and lack of belief in problem solving do not enhance students’ problem solving skills. Teachers who have wrong conceptions will therefore be sensitized to the need for modifying their behavior. The result of this study also indicates the relative pattern of interaction in mathematics classroom and provides recommendations that may stimulate teachers to improve on their teaching behavior to maximize student learning.
Students will benefit from the findings of this study because when they are taught by teachers who have modified their behaviors especially with regards to classroom interaction pattern, their conceptions about problem solving, and their firm belief in it; there is every possibility that they will acquire good problem solving skills that could enhance their performance in mathematics.
The result of this study will be beneficial to researchers. It would provide valuable information for those of them who are in search of solutions to students’ poor performance in mathematics and to the improvement of mathematics teaching and learning. The result of the study would serve as a foundation for further studies. The result of the study would act as a baseline for other researchers to continue investigations of teachers’ classroom instruction with regards to problem solving not only in secondary schools but at all other levels of mathematics studies.
Teacher educators will also find the result of this study helpful in developing teacher training programs for teachers. Problem solving strategies can be integrated in teacher training curriculum.
Textbook writers and curriculum planners will benefit from the results of this study. Information on what teachers know, think, and do in relation to problem solving, and thus could influence the direction of future curriculum development. And more often than not, school curricula and textbooks are agenda setters for school instructions and classroom activities. Curriculum planners and textbook writers using the result of this study would make better-informed decisions on content and approach, and incorporate problem solving strategies and frameworks in future mathematics textbooks.
Since problem solving skills help develop critical thinking and reasoning ability which are needed in many sectors of the society, it is also hoped that the wider society especially the business community and all those who use computations and figures, will benefit immensely from the results of this study.
Education policy makers will benefit from the result of this study especially when they recognize the deficiencies which mathematics teachers have and provide training on those deficiencies. It is then hoped that this study would generate enough steam for local and national workshops for teachers at all levels where teachers would be made to understand that problem solving is not an isolated part of the curriculum, but a rich part of what it means to do and understand mathematics.
Theoretically, the outcome of the study will provide information that will add to the existing postulations of theories related to problem solving especially with the cognitive learning theory, information processing and the principles of constructivism thereby contributing to the psychological understanding, philosophical explanation, and practical approaches to problem solving pedagogy.

Scope of the study

This study was limited to only exploratory and descriptive scopes of investigation as it aimed at unfolding and providing an insight on secondary school mathematics teachers’ conceptions and beliefs about mathematical problem solving. Two complementary data collection tools – survey questionnaire and classroom observation – were used.
With regard to geographical scope, Enugu and Anambra states were the research sites and secondary school mathematics teachers were the respondents. Secondary school level was used because teachers at this level are expected to have a minimum of NCE mathematics teaching qualification, which should be an indication of some exposure to mathematics pedagogy, and a possible reflection of mathematics instructional practice as well as have an opinion of what they teach and how they teach. This level is the one at which students develop whatever foundation that is laid in primary school, and it is a preparatory ground for career in mathematics and math-related professions; thus if wrong foundation was laid in primary school, that foundation can be corrected at the secondary level before students get to the university or go into various professions.
The content covered include: teacher’s conceptions of problem solving, teachers’ beliefs of problem solving, problem solving strategies/skills, instructional management setting, types of problems presented, teaching experience, and classroom interaction patterns.

Research questions and hypotheses
There are seven research questions and eight null hypotheses that addressed issues of teachers’ conceptions and beliefs about problem solving, the status of problem solving instruction, and the types of problems teachers present in class, among others.

Research questions
The following are the seven research questions which this work attempted to answer.
1. What are secondary school teachers’ conceptions of mathematical problem solving?
2. What are secondary school teachers’ beliefs about mathematical problem solving?
3. What is the status of problem solving instruction in secondary schools?
4. What problem solving strategies do teachers employ when teaching mathematics or solving problems for students?
5. What type of problems do teachers present during mathematics instructions or during problem solving?
6. What is the nature of interaction patterns in a typical secondary school mathematics classroom?
7. What is the nature of instructional pattern in a typical secondary school mathematics classroom?
Research hypotheses
Eight null hypotheses which guided this work were generated and tested at α ≤ .05. These were:
1. There is no significant relationship between the scores on mathematics teachers’ conceptions and beliefs about mathematical problem solving and what they teach.
2. There is no significant relationship between the scores on mathematics teachers’ conceptions and beliefs about mathematical problem solving and how they teach.
3. There is no significant difference among teachers’ mean conception of mathematical problem solving across teaching experience levels.
4. There is no significant difference among teachers’ mean beliefs about mathematical problem solving across teaching experience levels.
5. There is no significant difference between the mean conceptions of mathematical problem solving of male and female mathematics teachers.
6. There is no significant difference between the means of male and female mathematics teachers in terms of their beliefs about mathematical problem solving.
7. The relationship between the proportion of teachers’ responses on conceptions of problem solving and their academic qualifications is not significant.
There is no significant relationship between the proportion of teachers’ responses on belief about problem solving and academic qualifications

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