One of the major concerns in mathematics curriculum in Malaysia, especially in the secondary school setting, is to enable students to become as effective mathematical problem solvers.
"Students will apply mathematical concepts and skills and the relationships among them to solve problem situations of varying complexities. Students also will recognize and create problems from real data and situations with and outside mathematics and to apply appropriate strategies to find an acceptable solution. To accomplish this goal, students will need to develop a repertoire of skills, heuristics and strategies for solving a variety of problem types." (Curriculum Development Centre, 2001)
Problem solving plays a fundamental part in the learning and understanding of mathematics (e.g. Polya, 1973; Kilpatrick, 1987; NCTM, 2000). NCTM (2000), for example, defined problem solving as "engaging in a task for which the solution method is not known in advance," and suggested that "solving problem is not only a goal of learning mathematics but also a major means of doing so" (p. 52). It was found that what might be a significant task for one student could be routine or second-nature for another (Schoenfeld, 1985). Thus, the mathematics teacher should be challenged to pick a quality problem whose solution strategies are not immediately known to each student, but which is within each student's grasp (Perrin, 2007).
In order to provide a quality problem to each student, I would propose that students should be involved in problem posing environments. As researchers (e.g., Kilpatrick, 1987; Cai, 1998; English, 1998; NCTM, 2000; Doerr & English, 2003; Cunningham, 2004; Christou, Mousoulides, Pittalis, Pitta, & Sriraman, 2005 Tchoshanov, 2006) asserted that problem posing could help to develop students' problem solving ability. In a point of views, problem posing can be defined as a generation of new problems or a reformulation of given problems.
It was found that there are only a few related studies (e.g., Rohana, Munirah & Ayminsyadora, 2009) about problem posing performed in our country with regard to primary school students. Hence, I intend to shift the focus on secondary school students' abilities via their involvement in problem posing tasks/activities in our country.
It is important to note that mathematics teacher in Malaysia should also emphasize on problem posing tasks, instead of problem solving tasks alone (e.g., Siow, Hamzah & Chua, 2005). Watson and Mason (2002), for example, asserted that the shift of responsibility for problem posing from teachers to students could have a clear affective gain and necessarily involves some review of material used. Researchers (e.g., Xia, Lu & Wang, 2008; Cifarelli & Sheets, 2009; Priest, 2009) argued that by involving the activity of problem posing in mathematics classroom might help students in improving their problem solving abilities.
By considering these situations, problem posing tasks/activities should also be integrated and emphasized into the education system in our country. Thus, problem posing needs to be studied in our country.
In order to develop students' mathematical learning, thus the purpose of this study is first, to investigate the types of problem posing abilities among the middle secondary school students, as well as the upper secondary school students. Second, it tends to identify the types of problem posing difficulties, as well as the problem posing performances and the preferences of problem posing strategies among the middle secondary school students, as well as the upper secondary school students. Finally, it tries to develop problem posing tasks/activities which are relevant to the middle secondary school students, as well as the upper secondary school students.
1.2 Research Background
Problem posing has been regarded as the potential and important activities by previous researchers (e.g. Brown & Walter, 1993; Kilpatrick, 1987; NCTM, 2000) in mathematics education. It was stated that problem posing is a significant component of the mathematics curriculum and is considered to lie in the heart of mathematical activities. But it is still rarely emphasized in our country.
It is important for our mathematics curriculum to involve in problem posing environments. As Einstein and Infeld (1938) stated that:
"The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old questions from a new angle, require creative imagination and marks real advance in science." (Ellerton, 2000; Ellerton, Clements & Clarkson, 2000)
Researchers (e.g. Chin & Kayalvizhi, 2002; Bonotto, 2008) in mathematics education revealed that problem posing can (1) provide opportunities which stimulate higher order thinking by giving students carry out investigations, especially open-ended investigation, (2) foster a mindful approach towards realistic mathematical modeling and a problem posing attitude, (3) increase the problem solving achievement of success, (4) be a goal or a means of instruction, which involves asking students to respond to a range of problem posing prompts, (5) provide opportunities for the students to pose problems they enjoyed solving and promoted both a more complex and motivating learning environment, as well as (6) develop student's ability to identify and relate symbolic mathematical forms of knowledge to everyday life situations.
It is worthwhile for us to involve problem posing activities in mathematics education, instead of problem solving activities alone. Hence, the main purpose of the study is to investigate the types of problem posing abilities among the middle secondary school students, as well as the upper secondary school students. In addition, it also intends to identify the types of problem posing difficulties, the types of problem posing performances, the preferences of problem posing strategies, and to develop problem posing tasks which are relevant to the middle secondary school students, as well as the upper secondary school students.
1.3 Problem Statement
Reviewing the studies performed in our country, problem posing has long been under the shadow of problem solving in mathematics education. It is understandable that there is a close relationship between problem solving and problem posing. For example, problem posing can happen before, within, as well as after problem solving (e.g., English, 2003). Polya (1973) asserted that problem posing is crucial and included in his last stage of problem solving. For example, researchers are able to develop non Euclid Geometry (e.g., Hyperbolic Geometry and Elliptic Geometry) from Euclid Geometry (e.g., Euclid parallel axiom) (e.g., Bolyai & Lobachevsky, 1856; Riemann, 1866). Hence, it is important to involve students in problem posing tasks/activities as it can lead them to such kind of situations as being illustrated earlier. Also, there is a few studies about problem posing performed in our country (e.g., Ilfi, 2008; Ilfi & Md. Nor, 2009; Rohana, Munirah & Ayminsyadora, 2009), or there is some but not finished yet. Generally, problem posing can be divided into three main categories namely, opened problem posing tasks, semi-structured problem posing tasks, as well as structured problem posing tasks (e.g., Abu-Elwan, 2006; Ilfi & Md. Nor, 2009). Each of the problem posing tasks has their own characteristics respectively. For the purpose of this study, I would like to stress only on the structured problem posing tasks as it can help in shifting the autonomy of generating or reformulating new mathematics problems from teachers/textbooks to students. So, I intend to conduct a similar study regarding problem posing abilities in order to improve students' mathematical learning.
1.4 Research Objective
There are five purposes of the study, which to investigate the types of problem posing abilities, the types of problem posing difficulties, the types of problem posing performances, the preferences of problem posing strategies, as well as to develop problem posing tasks which are relevant to the two groups of students, namely the middle secondary school students, as well as the upper secondary school students. So, the objectives of the study are as follows, namely to:
investigate the types of problem posing abilities among the middle secondary school students, as well as the upper secondary school students.
identify the types of problem posing difficulties among the middle secondary school students, as well as the upper secondary school students.
identify the types of problem posing performances among the middle secondary school students, as well as the upper secondary school students.
identify the preferences of problem posing strategies among the middle secondary school students, as well as the upper secondary school students.
develop problem posing tasks which are relevant to the middle secondary school students, as well as the upper secondary school students.
1.5 Research Question
There is five research questions of the study that I would like to investigate as illustrated below:
What are the types of problem posing abilities among the middle secondary school students, as well as the upper secondary school students?
What are the types of problem posing difficulties among the middle secondary school students, as well as the upper secondary school students?
What are the levels of problem posing performances among the middle secondary school students, as well as the upper secondary school students?
What are the preferences of problem posing strategies among the middle secondary school students, as well as the upper secondary school students?
What types of problem posing tasks are relevant to middle secondary school students, as well as the upper secondary school students?
My suspicion was that students had little problem posing abilities and that this accounted for the poor performances on problem posing tasks. However, I wished to investigate specific areas of their problem posing difficulties and their preferences of problem posing strategies (or lack thereof) in order to develop appropriate problem posing tasks relevant to the middle secondary school students, as well as the upper secondary school students.
1.6 Theoretical Framework
The study involves the use of social constructivism theory (Ernest, 1991) and inquiry based learning (Bruner, 1966) as its theoretical framework. It consists of the integration of three kinds of learning theories namely, Dewey's learning theory (1997), as well as Vygotsky's learning theory (1978). Table 1.5 below shows the theoretical framework for the study.
1.6.1 Social Constructivism Theory
Ernest (1991) viewed mathematics as a social construction. This is aligned with the philosophy of mathematics of social constructivism. This philosophy regarded "human language, rules and agreement play a role in establishing and justifying the truths of mathematics" (p. 42). He classified this philosophy into three types, namely, first, linguistic knowledge, conventions and rules form the basis for mathematical knowledge, second, interpersonal social processes are needed to turn an individual's subjective mathematical knowledge into accepted objective knowledge, and lastly, objectivity is understood to be social. In other words, a key component of what separates social constructivism from other philosophies of mathematics is that it takes into account the interplay between subjective and objective knowledge. For example, subjective knowledge later becomes objective knowledge or knowledge accepted by the community when a discovery is made by an individual. On the other hand, this objective knowledge becomes subjective again as this knowledge is further spread to others and they internalize it.
In order to relate the above statements to the study, I would like to give a brief explanation regarding how social constructivism theory is related to it. To encounter problem posing tasks/activities, for example, students should first understand the given problem posing situations (e.g., opened problem posing situation, semi-structured problem posing situation, and structured problem posing situation). Second, they are given an opportunity to discuss (e.g., using language) with their peers (e.g., interpersonal relationship) regarding the meaning of such problem posing situations. Third, at the end of the discussion, they are required to pose their own problems respectively. Fourth, the posed problems would be encountered by their peers (e.g., to identify the objectivity of the knowledge) in order to ensure that whether it is solvable problems (e.g., objective knowledge) or unsolvable problems (e.g., subjective knowledge). Lastly, if the posed problems are unsolvable problems (e.g., subjective knowledge), the teacher would then guide the students (e.g., interpersonal relationship) in order to alter the problems so that it would become as solvable problems (e.g., objective knowledge). Hence, this study would regard social constructivism theory as one of its theoretical framework.
1.6.2 Inquiry Based Learning
Inquiry based learning or Inquiry based science referred to a range of philosophical, curricular and pedagogical approaches to teaching. Its main idea is that learning should be based around student's questions. This means that pedagogy and curriculum should require students to work together to solve problems. Therefore, the teacher's job in an inquiry based learning environment is to help the students along the process of discovering the knowledge themselves. In this form of instruction, teachers should be viewed as facilitators of learning. As being illustrated by Bruner (1966), inquiry based learning is a form of active learning, where progress is assessed by how well students develop experimental and analytical skills rather than how much knowledge they possess.
On one hand, Bruner (1966) asserted that there are three types of representation of human knowledge in mathematics, namely, Enactive, Iconic and Symbolic, and on the other, Tall (1997, 2003) stressed that there are several types of symbolic representation, for example, verbal (language, description), formal (logic, definition) and proceptual (numeric, algebraic). Figure 1.1 reveals an illustration of Bruner's theory (1966).
Figure 1.1: Three worlds of mathematical thinking
1.6.3 Face To Face Classroom
Optimal learning and human development and growth occur when people are confronted with substantive, real problems to solve (Dewey, 1997). According to him, curriculum and instruction should be based on integrated, community based tasks and activities that engage learners in a form of pragmatic social action that have real value in the world. Figure 1.2 reveals an illustration of Dewey's theory (1997).
Figure 1.2: Dewey's learning theory (1997)
Vygotsky's learning theory (1978) regarded teacher as an expert. He believed that cognitive development is the product of social and cultural interaction around the development and use of tools of a cognitive, linguistic and physical nature. Learning occurs in a Zone of Proximal Development when teachers act as mentors initiate and lead students as novices into problem posing environments. This structured introduction into using problem posing approaches is called scaffolding. In other words, works should be structured around projects that demand students engage in the solution of a particular community based, school based or regional problem of significance and relevance to their worlds. Figure 1.3 reveals Vygotsky's learning theory (1978).
Figure 1.3: Vygotsky's learning theory (1978)
In the face to face learning environment, the "didactic triangle" represents a model with the multiple relations among the three vertices namely, the student, the teacher and the mathematics (e.g. problem posing tasks/activities) as illustrated in Figure 1.4 (Tall, 1985; Albano, 2005). Therefore, the model and strategy that are used to promote problem posing must identify the role of each components and relations among them.
Figure 1.4: The didactical triangle
Problem posing is a way of promoting mathematical thinking among students in mathematics education. Several researchers (e.g. Mason, 1982; Tall, 1993) asserted that mathematical thinking is a dynamic process which expands our understanding with highly complex activities, such as abstracting, problem solving, conjecturing generalizing, reasoning, deducting, and inducting. As being illustrated earlier, Tall (2004), for example, categorized mathematical thinking into three significantly worlds namely, conceptual-embodied world, proceptual-symbolic world, and axiomatic-formal world. The theory of three worlds of mathematical thinking provides a rich structure in which to understand and interpret mathematical learning and thinking at all levels (Tall, 2007):
The conceptual-embodied world based on our physical perceptions that are built into mental conceptions through reflection and thought experiment.
The proceptual-symbolic world that begins with real-world actions (e.g. differentiation, integration) and symbolization into concepts (e.g. derivative, integral) developing symbolic that operate both as processes (e.g. differentiation, integration) to do and concepts (e.g. derivative, integral) to think about (called precept).
The axiomatic-formal world based on formal definitions and proof (Tall, 2004; Tall, 2007).
Several researchers (e.g. Roselainy, Sabariah & Yudariah, 2007; Sabariah, Yudariah & Roselainy, 2008) had used themes and mathematical processes through especially designed prompts and questions to cultivate and support students' use of their own mathematical thinking powers during face to face (FTF) interactions in classroom settings. Their model is based on cultivating students' mathematical powers in enhancing their problem solving skills, and promoting soft (generic) skills that can contribute towards students' acquirement of necessary attributes in learning mathematics. The entire theoretical framework for the study is shown in Figure 1.5.
Figure 1.5: Theoretical framework for the study
1.6.4 Problem Posing Concept
The study involves the use of "posing mathematical problems from given textbook problems" as its problem posing activities (Abu-Elwan, 1999, 2006; Ilfi, 2008; Ilfi & Md. Nor, 2009). According to this view, there are two phases in the solution process during which new problems can be created, the solver can intentionally change some or all of the problem conditions to see what new problem might result and after a problem has been solved, the solver can look back to see how the solution might be affected by various modifications in the problem.
Students who involve in this type of activities must write a description of each step as follow.
Choose a problem from mathematics textbook or mathematics workbook of secondary school (namely, Form Four or Form Six) settings.
Determine its conditions and its unknown by asking questions such as:
What is the problem all about?
What am I givens and non-givens?
What do I need to find?
Solve the problem via the suitable problem solving strategy.
Change the problem conditions in two different ways using "What if-not" strategy (Lavy & Bershadsky, 2003; Ilfi, 2008,; Ilfi & Md. Nor, 2009):
Students may use any of the following techniques in writing a new related problem:
Change the values of the given data,
Change the context,
Change the number of conditions.
Figure 1.6 reveals a modification of Abu-Elwan's (2002) framework of cyclic of activity problem (solving-posing).
Figure 1.6: Problem posing framework
1.7 Research Rationale
This section describes with regard to one major strand of research rationales. It describes about rationale of selecting problem posing tasks or activities:
Problem posing tasks become as a powerful assessment tool (e.g., English, 2003, Nicolaou & Philippou, 2007), which can reveal much about the understandings, skills and attitudes the problem poser brings to a given situation.
Problem posing is linked to metacognitive thoughts, so it could be adopted to differentiate between "more able" students and "less able" students in mathematics (e.g., Lowrie, 2002).
Problem posing is a way of contextualizing problems, for example, writing problems for friends to solve (e.g., Ellerton, 1986; Stoyanova, 1998; Lowrie, 1999, 2000, 2002) requires reflection and careful planning.
Problem posing helps to develop students' mathematical thinking (Abu-Elwan, 1999; Stoyanova, 2000, 2003; Ilfi, 2008).
Problem posing assists to improve problem solving performance (e.g., Lowrie, 2002; Abu-Elwan, 2006).
Problem posing is important for students' psychological and intellectual development (Rizvi, 2004).
Problem posing gives students autonomy of creating their own learning (NCTM, 2000).
Problem posing assesses students at the second highest level of Bloom's taxonomy (Bloom et al., 1956; Kastberg, 2003), namely synthesis level.
Problem posing as a window into students' understanding of mathematics, as a way to improve students' disposition towards mathematics, and as a way to help students become autonomous learners (Dickerson, 1999).
1.8 Research Significance
The findings of the study could be used to help the educators in developing a meaningful learning environment especially appropriate problem posing tasks or activities to their students.
The findings could help the students in getting an instant feedback regarding their abilities and performances in the problem posing activities. It is hoped that the students would be able to improve their abilities and difficulties, as well as their performances and strategies in the problem posing tasks or activities accordingly.
The teacher educational institutions could also get benefits from the findings. This could lead to the selection of the right candidates who would like to pursue their studies in mathematics education fields. This means that it could help us in generating prospective teachers who are not only capable in solving problems effectively, but also as creative problem posers.
Finally, the Ministry of Education (MOE) could design a suitable action based on the related findings. This could allow the authority in building a meaningful environment to students, as well as to provide several teaching strategies which are related to problem posing to teachers.
1.9 Research Scope
The major purpose of this study is to identify students' problem posing abilities at different levels. This study involves students in Form Four, as well as Form Six of secondary school settings. The results will be given based on three types of data collection techniques, namely students' written responses of test sessions, field notes of classroom observations and students' transcription of interview sessions. The selection of samples who involves in the interview sessions will be based on the Malaysian public examination results, namely PMR, as well as SPM. The researcher is confident with the cooperation and active involvement from the subjects, as all of them are volunteers of the study.
1.10 Definition of Terms
The following illustrates each definition of terms used in the study.
1.10.1 Problem Posing Tasks or Activities
The study adopts a modification of problem posing tasks or activities, which had been included in Abu-Elwan's (1999, 2006), as well as Ilfi's (2008) study. Problem posing tasks are viewed as the creation of a new problem which includes two phases in the solution process which new problems can be created, namely the solver can intentionally change some or all of the problem conditions to see what new problem might result and after a problem has been solved, the solver can look back to see how the solution might be affected by various modifications in the problem.
1.10.2 Types of Problem Posing Abilities
"Types of problem posing abilities" for the study refer to students' abilities in generating new problems, after solving a given "Original textbook problems" (Abu-Elwan, 1999, 2006; Ilfi, 2008; Ilfi & Md. Nor, 2009). First of all, the posed problems would be classified into two main categories, namely "Changing of the problem question" and "Changing one of the data problem" as shown in Figure 1.7. And, the change of the data problem could lead to two main types of the posed problems, namely solvable problems and unsolvable problems (Figure 1.8).
Figure 1.7: Hierarchy of the offered categorization according to data and question change by the students (Lavy & Bershadsky, 2003)
Figure 1.8: Hierarchy of the offered categorization according to data influence on the problem solution (Lavy and Bershadsky, 2003)
1.10.3 Types of Problem Posing Difficulties
"Types of problem posing difficulties" for the study refer to the types of difficulties when students encounter the problem posing tasks or activities, for example, the types of difficulties when posing new problems related to real life situations (e.g., Bonotto, 2006, 2007), as well as when posing problems using problem posing strategies, namely "What if-not" strategy (e.g., Lavy & Bershadsky, 2003; Lavy & Shriki, 2007).
1.10.4 Levels of Problem Posing Performances
"Levels of problem posing performances" in this study refer to the ability of completing the "solving problem", as well as the "posing problem" for the given questions correctly. This means that students should receive a total rating of "10" for the given questions in order to be classified as the top performer.
1.10.5 Preferences of Problem Posing Strategies
"Preferences of problem posing strategies" for the study refer to "What if-not" strategy (Abu-Elwan, 2002, 2006; Ilfi, 2008; Ilfi & Md. Nor, 2009), especially imitation strategy. The imitation strategy takes into account two important issues, the problem posing product has an extended structure and the student has encountered these types of problems (e.g., Stoyanova, 2005; Kojima, Miwa and Matsui, 2009). In other words, a problem posing strategy will be referred to as imitation when the problem posing product is obtained from the given problem posing prompt by the addition of a structure which is relevant to the problem, and the problem posing product resembles a previously encountered or solved problem. For the purpose of this study, imitation strategy refers to two types of problem posing techniques, namely "change the values of the given data" and "change the context" respectively (e.g., Abu-Elwan, 2002, 2006; Ilfi, 2008; Ilfi & Md. Nor, 2009). An illustration of "What if-not" strategy is shown in Figure 1.9.
Figure 1.9: Schematic description of "What if-Not" strategy (Brown & Walter, 2005)