Problem posing is commonly perceived as a cognitive activity that emerges in the process solving a problem but appears less commonly in the process of classroom instruction. The creation and reformulation of mathematics problems engages students’ thinking and their inquisitiveness in mathematical learning. This dissertation consists of three articles that explore the potential of problem posing for improving the teaching and learning of mathematics. This dissertation study begins with a meta-analytic study of research findings on classroom instruction based on problem posing activities. The Hedges’ g effect size is utilized to measure the effect of the problem posing instruction from 13 published studies. Four learning outcomes are identified from the studies: mathematics achievement, problem solving skills, levels of problems posed, and attitudes toward mathematics. The second article focuses on finding the relationship between problem posing and problem solving. Fifty one middle school preservice teachers participated in this study and completed two tasks related to problem posing and problem solving. Rubrics were developed to assess the written responses that revealed participants’ abilities in solving and posing mathematical problems. A fully mixed methods research design is utilized in article three for examining the effect of a fraction instruction on the level of elementary preservice teachers’ content knowledge, pedagogical content knowledge, and attitudes towards fractions. The instruction focused on using concrete models, problem solving, and problem posing activities for developing elementary preservice teachers’ knowledge of teaching fractions. The results from these studies revealed that problem posing is an effective approach as an inquiry-based instruction for improving students’ learning in mathematics. Research efforts are needed to further the type of studies that can provide teachers with specific approaches in developing and using problem posing strategies in the mathematics classroom.

This study discusses third grade elementary school children¡¦s growth in mathematics ability through a mathematics curriculum integrated with problem posing. By applying action research method, this study tries to find out the participating teacher¡¦s progress in her teaching method. In particular, also to find out the problems that may arise from carrying out problem- posing instruction, the possible solutions to these problems, and the problem-posing activities that can be readily applied in classroom setting. The time this study takes, from identifying the problem to the point of retreating from the research setting, is about one year. During this year, the investigator used various methods to collect data: interviews, video recordings of class teaching, teachers¡¦ notes and recordings, and children mathematics diaries. He included research-related surveys and triangulation for data analysis. It is found that, the teacher has to do more than having a firm grasp of the teaching materials, but also to attend to the teaching processes, the timing of introducing the critical questions related to the main points, and the ability to conduct a discussion atmosphere that is conducive for the students¡¦ growth. Most important of all, the mindset has to be changed from a teacher-oriented one to a more student-oriented set. Compared to the original teaching without problem posing, it is found that students became more active in class discussions, and showed more logic in their solutions. They debated about mathematics problems instead of aimless quarreling. Moreover, the content of the problem-posing products become more creative and no longer mechanical. In addition, students¡¦ solution processes were no longer by imitation, but instead, showing expressions of elaborated thoughts. All these show that the integration of problem-posing instruction has increased students¡¦ interests and motivation in studying mathematics.

The main purpose of this research is to explore the implementation of problem-posing teaching activities for fifth grade students in the elementary school. The teaching material is on mixed operations of addition, subtraction, multiplication and division. The method of posing problems is Tsubota¡¦s ¡§Classified Subject¡¨, adopted from Japan. The teaching of posing problems was divided into two phases; one is ¡§problem solving¡¨, the other is ¡§problem posing then solving¡¨. According to this method, students initially solve the problems that the teacher provided. Second, taking this subject as the foundation, students posed the problems by themselves and solved the problems as well. During this research, the researcher utilized a variety of ways to collect data, such as self-construction of instruments on four arithmetic operations, problem-solving worksheets, problem-posing worksheets, learning diaries, and reflective notes. The goals of this research are four: first, analyzing the categories of students¡¦ work and the contents of posing problems that student created; second, investigating into the performance of problem solving; third, probing students¡¦ opinions of problem-posing activities; four, the difficulties the teacher encountered. The results of this research were four. First, it showed that 98.5% of students given problems included sufficient data for solving. Students virtually were able to make feasible problems. Moreover, the majority of students were capable to, not only changing numerals of the problems, but also changing structures of the problems. The tendency of changing structure followed multiple aspects of developments. Second, students¡¦ performance on three steps operations problem solving was low; the performances of problem solving and problem posing then solving were close; students¡¦ performance at problem posing then solving stage was higher; and, the major reason for mistakes was insufficient procedural knowledge. Third, students expressed a liking of problem posing, they thought that the materials were interesting, and showed promising study manner. Fourth, the teacher encountered problems such as time control, the development of in-class presentation culture, and, few students¡¦ lack of concentration while problem posing.

The purpose of this study is to investigate the implementation of problem-posing instruction on fraction in a fifth-grade elementary school of aboriginal children in Taitung. Through a pilot and integrating modified problem-posing instruction in mathematics teaching, the investigator studied the performance and learning attitude of children and analyzed the acceptability of the problem-posing teaching processes. The researcher collected data by using: own constructed fraction problems question sheet, worksheet on problem-posing, worksheet on problem-solving, the teacher¡¦s math notes on instruction, children¡¦s diaries, students¡¦ feedback surveys and post-tests of mathematical problem-solving ability. The researcher analyzed categories of children¡¦s work and contents of problems-posing that children created. Results indicated that the children made progress in problem-posing performance and ability of problem-solving and behaved positively on learning attitude. From this study, the researcher found that the majority of the students participated in this study were interested in this teaching technique, and students gained confidence in posing and solving mathematical problems. Finally, the teacher could reflect upon practice on problem-posing instruction through action research. The above results yielded instructional implications for teachers who consider integrating problem-posing teaching into mathematics instruction for elementary school children.

The purpose of this research is to explore the implementation of problem posing teaching activities in the seventh grade math class, including cooperative posing and individual posing, and to suggest specific teaching methods to those teachers who are interested in introducing problem-posing instruction in their classes. The research subjects were from one of grade seven classes and class materials were mainly based on textbook. In the first semester of the school year 2002, students received in a traditional mathematics class, and then in the second semester, they received one problem-posing lesson per week self-study period. There are two phases for this problem-posing research: four times cooperative posing and three times individual posing. During this research period, the researcher used a variety of ways to collect data, such as observing, interviewing, video-taping, self-introspecting, and asking students to keep diaries. The researcher examined results by triangulation and evaluated students¡¦ problem-posing abilities. The result of this research showed that students performed differently in different units. Of the seven units, the order of the highest score to the lowest is: Negative numbers, Volume and capacity, Approximation, Division of fraction, H.C.F. and L.C.M., The four basic operation, and, Number and Measures. In this regard, the researcher suggested that if teachers want to integrate problem posing into instruction , it would be more appropriate to apply to those units students received higher scores. As for the early phase of this implememtation, students did not know how to discuss with each other. Gradually they improved and understood the meaning of team work. As for the topics of activities, some students came out with something related to names and life events; other students used news and adolescent topics as discussion materials. As for evaluating classmates¡¦ topics, students did not know how to give suggestions nor to spot other classmates¡¦ mistakes. Sometime, they contradicted themselves when they gave suggestions. Finally they could focus on data, discussed, and gave concrete suggestions. The researcher also found that students evaluated the numerical information content of the problems they posed and checked if they are reasonable and if the problems meet teachers¡¦ requirements. As for editing their own questions, some students did not pay attention to their classmates¡¦ suggestions; some paid attention to peer suggestions but made the problems worse. After thorough practice, students learned how to make proper revisions. In all, there are advantages of implementing problem posing into matehmatics instruction. The advantage of cooperative posing is to create a team learning environment while the advantage of individual posing is to stimulate individual creative thinking in posing problems.

The aim of this study is to investigate differences between aboriginal and Han fifth graders in problem posing. The objectives of research include: (1) design and implementation of arithmetic problem posing process; (2) investigation of the arithmetic learning condition for aboriginal and Han pupils; and (3) investigation of the misconception presentation difference between aboriginal and Han pupils in problem posing. The problem posing process falls into 3 stages: ¡§problem solving¡¨, ¡§problem posing¡¨, and ¡§problem posing-solving¡¨. This study applied the problem solving and problem posing learning sheets for pupils to engage in individual problem solving and problem posing. Pupils were requested to complete a learning journal after class. As a teacher, the researcher allowed aboriginal and Han pupils to engage in problem solving and problem posing based on 3 types of questions: word algorithm, calculation, and open-end questions; and collected data with learning journals and pupil interviews. Findings include: (1) pupils have different performances when posing different types of problems; (2) both aboriginal and Han pupils can pose feasible and appropriate questions, particularly for word items, though it is not easy for pupils to pose appropriate open-end questions; and (3) the content of problems posed varies as a result of environmental and cultural differences. Additionally, the problem posing teaching allows pupils to feel more interested in learning mathematics, and such a positive learning attitude can enrich the mathematic concepts and enhance the thinking ability of pupils. Keywords: problem posing, four basic operations, aboriginal children

The purpose of this study is to integrate problem-posing teaching activities into mathematics curriculum for grade four elementary school students. The contents include three units on number and operation. Data collection included using the parent-child problem-posing worksheet, student mathematics diary, teacher teaching journal, mathematics attitude measurement form, parent and students interviews, in order to study the influence of problem-posing teaching on teacher and students mathematics attitude and parent-child interactions. The investigator aimed at bringing up specific suggestion on teaching. There are six results in this study: 1. Problem-posing teaching can increase students¡¦ problem-posing ability. 2. Problem-posing teaching can improve students¡¦ mathematics attitude. 3. Problem-posing teaching can enhance students¡¦ mathematics learning. 4. Parent-child problem-posing can promote parent-child co-learning, and increase parent-child interactions. 5. Parent-child problem-posing can make parent understand their children¡¦s mathematics curriculum, and they are not afraid when supervising children. 6. In teacher¡¦s reflection, it recorded that a change of teaching mode to problem posing to promote mathematics learning was feasible, and problem-posing teaching fit in the ¡§student-centered¡¨ teaching spirit of Nine-years-integration.

Research study on sixth grade problem-posing instruction: Case of addition, subtraction and number comparison on decimals Abstract The aim of this research project is to investigate the implementation of problem-posing instruction on decimals to one sixth-grade mathematics class. There are four research objectives: 1) design and implement problem-posing instruction on decimals; 2) discuss the status of children¡¦s performance in problem-solving; 3) analyze the type of problems posed by children; and, 4) display categories of misconceptions exhibited when children did problem posing. The stages for instructions were three: 1) children solved the problem given by the instructor; 2) children referred to given problem and posed a problem; and, 3) children solved their own problem. In this study, the type of problem posing chosen for instruction is ¡§similar problem¡¨, which is adapted from Tsubota, a Japan scholar. The researcher collected data by using: own constructed decimal problems question sheet, worksheet on problem solving, worksheet on problem posing, children¡¦s diaries and teachers¡¦ notes on instruction. There are four findings. First, the implementation of sixth grade problem-posing instruction on decimals is feasible. Second, 96.9% of students¡¦ problems are plausible and contain sufficient information for problem solvers. Most students could change the number and content of the question but few revised the structure of the question. There was also multiple development for those problems. Third, children¡¦s performance in posing/solving stage was better than that in problem-solving stage. Finally, the researcher reported that the teacher faced problems such as difficulty in control of time, establishing children¡¦s habit in reporting, and collecting misconceptions of children. Key word : problem solving; problem posing; addition, subtraction and number comparison on decimals

The researcher used self-constructed problem-solving and problem-posing activity sheets to ask students to do solving and posing by themselves, then discuss together afterwards. When teaching was completed each time, the researcher asked students to write learning diaries. The researcher referred to the students¡¦ records on problem-solving and on problem-posing and analyzed the categories of students¡¦ work and the contents of posing problems that students created as well as problem-solving strategies. He also referred to students¡¦ learning diaries to investigate into students¡¦ reflections on their problem-posing, and to make a record of self reflections during his problem-posing activities implementation. The results of this research showed that most of the students given problems included sufficient data for problem solving. Students virtually were able to make feasible problems but only few students can change the structures of the original given problems. Most of these problems are ¡¥Exchange problem¡¦ which be solved correctly by themselves. Most students solved problems by using the strategy relating to Multiples. The most important factor for accuracy and solving strategy is number type on the proportion problems. Moreover, students expressed that they were able to acquire study methods and have great enjoyment from problem-posing. Also, the teacher found that during problem-posing, students will advance in thinking and creativity.

The main purpose of this study is to understand the possible effects of the problem-posing teaching technique in an elementary school, grade-two class. With problem-posing teaching technique of twenty-four classes within 12 weeks, the experimenter first used the problem-posing texts to ask students to formulate mathematical problems. After reviewing the problems formulated by students, the necessary interviews were done. The statistical analysis is done on pre-tests and post-tests of mathematical-solving ability. Students¡¦ feedbacks about problem posing teaching technique are collected. There were four stages in this experiment: (1) oral presentation of problem-posing, (2) written presentation of problem-posing, (3) written presentation of problem-posing and problem-solving by the same person, (4) wiitten presentation of problem-posing by one and problem-solving by another. The experimenter explored the following themes during these four stages: (1) the process of problem-posing learning, (2) the characteristics and erroneous types of the students¡¦ opus, (3) the differences on the problem-posing abilities when students faced formulas, pictures and written contexts, (4) the behaviors of the high problem-posing ability group and the low problem-posing ability group, (5) the enhance of problem-solving abilities due to problem-posing teaching technique. From this study, the experimenter found that the majority of the students participated in this study interesting in this teaching technique, and students¡¦ gained confidence in posing and solving mathematical problems. Besides, the experimenter also found that: (1) the students¡¦ ability in posing and solving problems progressed gradually in speed and correctness, (2) the characteristics and erroneous types of the students¡¦ opus were diverse, which included relativeness of situations, students¡¦ interests, school lives, and daily lives, correctness of mathematical logic, ambiguity of language, (3) students were better to pose problems from pictures and written contexts than from formulas, but there was no difference between from pictures and from written contexts, (4) the high problem-posing ability group performs better in speed and correctness to solve problem than the low problem-posing ability group, (5) Comparing the controlled and non-controlled groups, problem-posing teaching technique seems to helpe students to enhance their problem-solving ability.