Discourse

Another component of PBL that facilitates mathematical thinking of students in the classroom is known as discourse, that is, teachers’ ability to create an environment that makes students forget Mathematics anxiety. According to Reys & Long (1995), the discourse of a classroom, that is, the ways of representing, thinking, talking, agreeing and disagreeing – is central to what students learn about Mathematics as a domain of human inquiry with characteristic ways of knowing. Discourse is both the way ideas are exchanged and what the ideas entail. Students must talk with one another as well as in response to the teacher. When the teacher talks most, the flow of ideas and knowledge is primarily from teacher to student. In relation to mathematical discourse, the teacher’s role is to translate what is being said into academic discourse, to help frame discussion, pose questions, suggest real-life connections, probe arguments and ask for evidence.

The language practices of the classroom (educational discourse) must ‘scaffold’ students’ entry into mathematical discourse (Adler, 1997). When students make public conjectures and reason with others about Mathematics, ideas and knowledge are developed collaboratively, revealing Mathematics as constructed by human beings within an intellectual community. King in Rosenshine & Meister (1992) reported that after hearing a lecture, students met in small groups and practiced generating questions about the lecture.

Students in Schoenfeld’s (1985) study had opportunities to participate in small group mathematical problem solving. He sequenced the problems he presented to his students when teaching mathematical problem solving. He first gave students problems that they were incapable of solving on their own; this provided the motivation for learning the strategy he planned to introduce. He suggests that small-group work facilitates the learning process in four ways. It provides support and assistance as students actively engage in problem solving and group decision making, facilitates the articulation of knowledge and reasoning as students justify group members reasons for choosing alternative solutions. Others are that students receive practice in collaboration, a skill required in real-life problem solving and

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students who are insecure about their abilities to solve problems have the opportunity to see more capable peers struggle over difficult problems (Rosenshine & Meister, 1992). The above scenario aptly distinguishes PBL environment from any other method of teaching for effective students’ learning. According to Wheijen (2005), the development of the constructivist view of learning in recent years has resulted in modifications of teaching design in many science classes.

2.10.4 Classroom Environment

A PBL classroom environment seems to be an antidote to students’ truancy and disruptive behaviour. According to Sungur & Tekkaya (2006) education research reveals that beliefs and cognition that enable students to be independent learners are related highly to academic learning. The viewpoint has led to an increased emphasis on how classroom context and other contextual factors shape and influence student learning and motivation. Educators, therefore focus their attention on students’ strategic efforts to manage achievement through specific beliefs and processes. According to Zimmerman in Sungur & Tekkaya (2006), those self- regulatory processes and beliefs had been the focus of systematic research. Constructivist teaching seemed to have guided the students towards coherent perceptions of constructivism including beliefs about effective learning and teaching strategies, epistemological beliefs about science knowledge, and perspectives on learning goals.

The outcomes of the constructivist teaching in developing students’ perspectives on how to learn and what to achieve as observed by Wheijen (2005) were in agreement with the studies of Elby (1999). No single theory is comprehensive enough to explain learning and, at the same time, reliably predict the best way to select and organise content and choose a teaching strategy. Few theories address such vital aspects of learning as affective behaviour and classroom climate. The belief of behaviourists and cognitivists on learning seems to have some variations. The common behaviourist’s definition of learning is that it is any change in behaviour (Green, 1968) or “the relatively permanent modification of behaviour as the result of experience” (Magee, 1971:71). This definition suggests that teachers should induce certain behaviours in students; when these behaviours are demonstrated, we can assume learning has taken place. On the contrary the position of cognitivists is that learning is not merely a change in behaviour.

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The cognitivists’ view behaviours as mere indicators of learning; learning itself is an internal process that takes place somewhere between the stimulus and the response. Certain behaviours might imply that learning has taken place, it may also have occurred when no overt behaviour change can be detected (Green, 1968:56). Teachers that have negative beliefs and mind set towards the workability of PBL will necessarily not implement or see any positive impact that PBL has in students’ proper understanding of mathematical concepts. 2.11 Conclusion

The conceptual analysis and theoretical framework of PBL had been discussed. The case studies that focused mainly on the first year undergraduate Mathematics courses as the origin of PBL were from a tertiary institution. Further Mathematics is not standing alone as a course or subject at tertiary level but it is embedded in the Mathematics courses. Its linkage with the secondary school level is because most of the Mathematics courses at the first year undergraduate level are contained in Senior Secondary School Further Mathematics curriculum for classes 1 to 3 in Nigeria. It is because of this reason that Further Mathematics is referred to as the bridge between Mathematics offered at secondary school and Mathematics courses at the first year undergraduate level at the tertiary level. The alternative methods to the Traditional method of which PBL is significantly one as could be seen in the conclusion foster on students’ understanding of Mathematics rather than the traditional method that is preoccupied with examination success and syllabus coverage. It should be noted however, that a student with proper understanding of any mathematical topic or concept would necessarily do well in any examination.

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CHAPTER THREE RESEARCH METHODOLOGY 3.1 Introduction

This chapter describes the methodology followed in addressing the research questions put forward to seek possible solutions to the problems identified in chapter one. In this section, research methodology/ paradigm, research design, population and sample, the research instruments, procedure for data collection, data analysis and interpretation, limitations of the study, and validity and reliability are discussed.