The influence of the learning environment upon knowledge development has received relatively little attention in the field of mathematics teaching and learning (Boaler, 1999; Samuelsson, 2008). Even so, teachers often expect researchers to provide that kind of knowledge in mathematics didactics.
What happens in the classroom has an impact on students’ opportunity to learn. The activities in the classroom, the repeated actions in which students and teachers engage as they learn are important because they constitute the knowledge that is produced (Cobb, 1998). There is some evidence that different teaching styles can have different impacts on student achievement (Aitkin & Zukovsky, 1994) and that the choice of teaching approaches can make an important difference in a student’s learning (Wentzel, 2002). The synthesis of meta-analysis and reviews of Teddlie and Reynolds (2000) gives evidence for positive relationships between achievement and varied classroom settings. Case (1996) argues that a variation of teaching methods is important because different teaching methods draw attention to different competencies in mathematics (e.g. Boaler, 2002; Samuelsson, 2008). Thus, the mode of teaching method in mathematics seems to be important for students’ development of mathematical proficiency.
There are very few studies focusing on how different teaching methods affect students’ calculation and conceptual understanding as well as self-regulated learning skills, but there are several studies that focus on closely related areas.
For learning in general, Granström (2006) shows that different teaching approaches in classrooms influence the outcomes for students in different ways. Settings where students are allowed and encouraged to cooperate with classmates and teachers give the students more opportunities to understand and succeed. Similarly, Oppendekker and Van Damme (2006) stress that good teaching involves communication and building relationships with students. Boaler (1999, 2002) reports that practices such as working through textbook exercises or discussing and using mathematical ideas were important vehicles for the development of flexible mathematical knowledge. One outcome of Boaler’s research was that students who had worked in textbooks performed well in similar textbook situations. However, these students found it difficult to use mathematics in open, applied or discussion-based situations. The students who had learned mathematics through group-based projects were more able to apply their knowledge in a range of situations. Boaler’s research gives evidence for the theory that context constructs the knowledge that is produced.
In a review of successful teaching of mathematics, Reynolds and Muijs (1999) discuss American as well as British research. A result of their review is that effective teaching is signified by a high number of opportunities to learn. Opportunity to learn is related to factors such as length of school day and year, and the amount of hours of mathematics classes. It is also related to the quality of classroom management, especially time-on-task. According to research in the area, achievement is improved when teachers create classrooms that include (a) substantial emphasis on academic instruction and students’ engagement in academic tasks (Brophy & Good, 1986; Griffin & Barnes, 1986; Lampert, 1988; Cooney, 1994), (b) whole-class instruction (Reynolds & Muijs, 1999), (c) effective question-answer and individual practices (Brophy, 1986; Brophy & Good, 1986; Borich, 1996), (d) minimal disruptive behaviour (Evertsson et al., 1980; Brophy & Good, 1986; Lampert, 1988; Secada, 1992), (e) high teacher expectations (Borich, 1996; Clarke, 1997), and (f) substantial feedback to students (Brophy, 1986; Brophy & Good, 1986; Borich, 1996). Aspects of successful teaching are found in a traditional classroom (lecturing and drill) with one big exception- in successful teaching, teachers are actively asking a lot of questions and students are involved in a class discussion. With the addition of active discussion, students are kept involved in the lesson and the teacher has a chance to continually monitor students’ understanding of the concept being taught.
On the other hand, negative relationships have also been found between teachers who spend a high proportion of time communicating with pupils individually and students’ achievement (Mortimer et al., 1988; OfSTED, 1996). Students’ mathematics performances were low when they practiced too much repetitive number work individually (OfSTED, 1996). A traditional direct-instruction/active teaching model seems to be more effective than a teaching model that focuses on independent work.
Another teaching model discussed in the literature is the one dependent on cooperative, small-group work. The advantage of problem-solving in small groups lies in the scaffolding process whereby students help each other advance in the Zone of Proximal Development (Vygotsky, 1934/1986). Giving and receiving help and explanations may widen students’ thinking skills, and verbalising can help students structure their thoughts (Leiken & Zaslavsky, 1997). The exchange of ideas may encourage students to engage in higher-order thinking (Becker & Selter, 1996). Students who work in small groups are developing an understanding of themselves and learning that others have both strengths and weaknesses. Programmes that have attempted problem-solving in small groups as a teaching method report good results, such as improved conceptual understanding and higher scores on problem-solving tasks (Goods & Gailbraith, 1996; Leiken & Zaslavsky, 1997).
Samuelsson (2008) used a split-plot factorial design with group (i.e., traditional, independent work, and problem-solving) as a between-subject factor and time (i.e., before and after a 10 week intervention) as a within-subject factor. In that design, traditional approach means that teacher explained methods and procedures from the chalk board at the start of the lessons, and the students then practice with textbook questions. Independent work means that students work individually on problems from a textbook without a teacher’s introduction to the lesson; teachers just helped students who asked for it. Problem solving means that students were introduced to different ideas and problems that could be investigated and solved using a range of mathematical methods. Students worked in groups of four, and they discussed and negotiated arithmetic issues with each other and with the teacher, both in groups and in whole-class discussions. There were a total of seven dependent variables in the study. There were three measures of mathematics abilities; that is, a total score of mathematics ability, calculation, and conceptual understanding. Measures related to self-regulated learning skills such as internal and instrumental motivation, self-concept, and anxiety were also used as dependent variables. The results showed that there are no significant interaction effects between group and time according to total arithmetic ability and calculation. However, differences in students’ progress in conceptual understanding may be explained by the teaching method. Traditional work as well as problem-solving seems to have more positive effects on students’ development of conceptual understanding than independent work does.
To develop aspects of self-regulated learning skills, teachers, according to Samuelsson (2008), would be advised to use traditional work or problem-solving. Problem solving appears to be more effective in developing students’ interest and enjoyment of mathematics than does traditional work or independent work. Also traditional work and problem-solving are more effective than independent work for students’ self-concept.
Thus, different teaching methods also seem to influence students’ self-regulated learning skills (interest, view of the subject’s importance, self-perception, and attribution) (Boaler, 2002). Students who were expected to cram for examinations describe their attitudes in passive and negative terms. Those who were invited to contribute with ideas and methods describe their attitudes in active and positive terms that were inconsistent with the identities they had previously developed in mathematics (Boaler, 2002). A negative attitude towards mathematics can be influenced, for instance, by too much individual practice (Tobias, 1987) as well as by teachers who reveal students’ inabilities. Students who do well in school (Chapman & Tunmer, 1997) demonstrate appropriate task-focused behaviour (Onatsu-Arvillomi & Nurmi, 2002), and they have positive learning strategies. If the students are reluctant in learning situations and avoid challenges, they normally show low achievement (Midgley & Urdan, 1995; Zuckerman, Kieffer, & Knee, 1998).
As a result, the choice of teaching method not only affects mathematics achievement but also students’ self-regulated learning skills.
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