How mentoring practises under problem posing instruction affects creativite thinking

Creativity is recognized as an important mathematical ability by mathematicians, educators and theorists. The study aimed that how to problem posing instruction has positive effects on the development of mathematical creative thinking under mentoring practices. 15 participants are selected randomly among prospective teachers from Suleyman Demirel University Mathematics Education Department. In the study, 15 word problems was prepared by using problem posing approach and activities according to this posing approach, that is especially open ended word problems, increased the verbal ability as it is one component of the creativity. Which is affected by studying the problem posing approach as a classroom activities.The research is quantitative and results are evaluated by SPSS t independent test. Results indicate that problem posing with mentoring activities increased the creative thinking for prospective teachers. The differences between pre and post tests are statistically significant. Problem posing activities with mentoring practices that includes personal attributes in the instruction on mathematics education has positive effects in the domain of creative thinking. 

LITERATURE REVIEW

The framework of the study was based on the five-factor mentoring model proposed by Hudson (2004) in the European Journal of Teacher Education. It illustrated explicit effectual practices within elementary science mentoring that promoted valuable teaching skills and dispositions within novice teachers. This form of mentoring enabled mentors to “become agents of systemic change” (p. 142). The five key factors when considering effective mentorship were personal attributes, system requirements, pedagogical knowledge, modeling and feedback.

Personal attributes underscored all subsequent factors because it required the mentor to develop a relationship with a mentee that was positive and supportive (Hudson, 2005). Mentors needed to demonstrate good listening skills, reflective discourse, and a willingness to pursue a mentee’s educational interests within the context of the classroom. Nested within personal attributes, the concept of educative mentoring illuminated mentors who assisted mentees “to interpret what their students said and did, and then to figure out how to move their students‟ learning forward” (Norman & Feiman-Nemser, 2005, p. 680). Educative mentoring was first described by Feiman-Nemser (1998) as “mentoring that helped novices learn to teach and develop skills and dispositions which encouraged continued learning in and from their practice” (p. 66). According to the five-factor model, system requirements pertained to obtainable goals for teaching elementary science, relevant school policies, and science content curriculum (Hudson, 2007; Shea & Greenwood, 2007). This involved the mentor relaying technical advice supporting elementary science instruction. The mentee was made aware of policies and practices for implementing curriculum documents including local safety concerns and issues. The next factor, pedagogical knowledge, involved content knowledge as well as planning, timetabling lessons, teaching strategies, problem solving, classroom management, questioning skills, implementing effective practices and assessment (Hudson, Skamp & Brooks, 2005; Schavarien & Cosgrove, 1997). Mentoring was a “think aloud intellectual activity” that assisted the mentee in “fostering an inquiry stance” toward developing “pedagogical thinking” about elementary science (Feiman-Nemser, 1998, p. 69).

Modeling, as the fourth factor, implied willingness by the mentor to model effective science instruction through enthusiasm, a rapport with students, lesson planning, syllabus language, hands-on lessons and classroom management (Appleton, 2008; Jarvis et al., 2001; Schavarien & Cosgrove, 1997). However, the modeling was a mutual activity in which the mentor and mentee each showcased individual instructional styles. The mentee engaged in the authentic task of teaching while the mentor took the lead when appropriate (Norman & FeimanNemser, 2005). Feedback, as the fifth factor, described a type of communication between mentor and mentee when evaluating the outcome of student learning and setting clear expectations for the mentee (Hudson, 2005). When practicing feedback in the form of educative mentoring, articulation was not a one-way street, but resulted from the mentor and mentee doing and talking about the work together (Feiman-Nemser, 1998).

Mathematician and educators recognizes mathematical creativity as a major element of mathematical ability and have tried to define it. After searching literature and research on mathematical creativity Aiken (1973) concluded that mathematical creativity is always defined on the basis of process and various products. Judging from literature and studies on mathematical creativity, the nature of creativity can be classified into two perspectives: Firstly, mathematical creativity is regarded as cognitive ability that leads to emphasize creative thinking. Secondly, mathematical creativity is essentially defined with focus on products. Generally mathematical creativity is based on fluency, flexibility and novelty (originality). Fluency refers to the number of ideas produced in responses to prompt; flexibility refers to the different approaches to the prompt; and the novelty is the originality of the ideas produced in response to a prompt.

Creative thinking in mathematics education is so important that’s why educators should find new methods to develop the verbal thinking ability in problem solving or to increase the ability of creativity. In this study, problem posing activities are designed to described situations in literature.

Brown and Walter (2005) stated that one of the important consequences in mathematics education is to provide opportunities to students in mathematics lessons for developing their problem posing skills. Because problem posing is not only to generate new problems from given situations but also reformulate given problem and generalize for the solution.

Silver (1994) proposed that problem posing has too much interest because of Its effect in creativity and mathematical ability, improving student problem solving ,a large window into understanding of mathematics.

Problem posing in contrast to traditional problem solving methods reduces anxiety and common fears about mathematics and increases positive attitudes toward mathematics (Philippou, Nicolaou 2004).

Silver (1997) argued that inquiry-oriented mathematics instruction which includes problem-solving and problem-posing tasks and activities can help the students to develop more creative approaches to mathematics. It is claimed that through the use of such tasks and activities, teachers can increase their students’ capacity with respect to the core dimensions of creativity, namely, fluency, flexibility, and originality (e.g., Presmeg, 1981; Torrance, 1988). English (1997a) claimed that in her study of a problem posing program, the activities had a strong emphasis on children being creative, divergent, and flexible in their thinking and students were encouraged to look beyond the basic meanings of mathematics with those activities.

Balka (1974) applied mathematical creativity that has three components as fluency, flexibility and novelty.he asked the questions to subjects to pose mathematical problems that could be answered on the basis of information provided in many stories taken from real world situations.

Getzels and Jackson (1962) asked problem posing tasks to the students to identify creative individuals. Mathematical creativity can be used as measuring factors. Fluency is used as measuring factor (Foster 1970; Baur 1971; Maxwell 1974; Dunn 1976). Second, flexibility is used (Krutetskii 1976). Third, fluency and originality are used (Mainville 1972). Fourth, fluency, flexibility and originality are used. (Evans 1964; Zosa 1978; Balka 1974; Kim 1997; Song 1998).

Problem posing, or problem finding, has long been viewed as a characteristic of creative activity or exceptional talent in many fields of human endeavor. For them, creativity is similar to problem posing in its multiplicity in nature for example, Getzels & Csikszentmihalyi (1976) studied artistic creativity and characterized problem finding as a center of creative artistic experience. Problem posing, along with problem solving, is central to the 166 Lee, Kang Sup; Hwang, Dong-Jou & Seo, Jong Jin discipline of mathematics and the nature of mathematical thinking ( Silver 1994). Guilford and his associates hypothesized that fluency,flexibility, and originality would be three important aspects of creativity (Guilford, 1959). Such traits were found in Guilford’s well-known structure of intellect model. Guilford claimed that the intellectual factors fall into two major groups–thinking and memory factors—and the great majority of them can be regarded as thinking factors. Guilford defined divergent production as the generation of information from given information that is nearly similar definition of problem posing. In the divergent thinking includes different unusual ideas produced by people. Fluency in thinking refers to the quantity of output. Flexibility in thinking refersto a change of some kind. Guilford saw creative thinking as clearlyInvolving what he categorized as divergent production.

For all educators, especially for us ,to evaluate the intelligence of any kinds of mathematical ability for the students, we use word problems to understand the level of them ,because we prepare the students for the future and we direct them for coming occupational positions.

By teaching the students word problems we extend their minds to think them creatively, to develop their problem solving abilities. By doing this, the students race themselves by solving many problems and result of making more cognitive practice.

The development of creative mathematical ability by problem posing is very common in literature also. Ill structured, open ended problems that are stated in a manner that permits the generation of multiple specific goals and possibly multiple correct solutions as a result of the students’ interpretation. For example Schoenfield (1985) asked “The Fermi style” problem of “how many cells are there in the body of an average adult male human?”. Silver (1994) states that these kinds of open ended problems give a rich source of experience in interpreting problems and perhaps generating different interpretations.

Results and Discussion

Results indicate that the average differences between pre-test and post test for experimental group is significantly high the average value of the pre-test result for experimental group is 14.55 but this results changes for post-test that is 17.52 .This shows that the difference is in a positive way. Therefore the interaction between student and teachers is more effective way to solve the questions and produce questions. But on the other hand this difference for control group does not give significant results. It means that traditional method means no more interaction in the class during problem posing and solving sections changes nothing.

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