Include the specific observations that stimulated your research proposal.
- From the Princess Elizabeth Primary School problem solving video, it was observed that the teacher regularly encouraged student participation and allowed students the freedom of expression. Even when students came up with the wrong concepts or incorrect answers, she did not put them down but offered them other alternatives and various process pathways to generate further thinking.
- The majority of the students displayed good enthusiasm and during the heterogeneous small-group discussions, they appeared to have good constructive interactions among themselves.
- It was also noted that when students did not obtain the correct answers to the questions posed by the teachers, they do not appear to be discouraged and yet showed perseverance to come up with alternative ideas to problem solve.
Your research proposal should include a statement of what you feel is productive to research further into.
For the lesson, it is essential that students have prior cognitive knowledge on the basic mathematical operations. However, as students go about their tasks, it was not easy to evaluate the difficulties and emotions experienced by the individual students and how they eventually react to any stalemates or deadlocks. Other than the cognitive domain, the affective traits could have imposed an influence on the development of the students’ behaviour and thinking. For example, students who were not participative may have believed that if they could not solve a problem in a few minutes, they would never find the solution and thus should not waste time on it. Such beliefs about mathematics shape students’ behaviour and can often produce negative consequences.
Thus, it is imperative that classroom teachers know what the different affective traits are, how they influence the students’ success in mathematical problem solving; and finally, how they can nurture the desired affective traits in the students. It was hoped that from the research, the role of affective traits in mathematical problem solving can be better understood, particularly in the field of proving mathematical statements.
Outline how you would go about to do this research.
Case studies will be done on two students within a Secondary 3 Express class of above-average ability. The class will be tasked to prove a particular mathematical statement and to indicate their proving procedures and the thoughts that accompanied the processes. The goal of this mathematical task is to make the students reflect on what happens when a proof is carried out and to develop skills in analyzing during proving processes. The written protocols of two students will then be examined.
These two case studies aim to illustrate that when it comes to mathematical problem solving, purely cognitive behaviour is extremely rare and that the cognitive and affective domains are intertwined. The expressions and quotes presented by the students will allow us to go deeper into the difficulties faced by the two students in terms of the lack of mathematical knowledge and skills and subsequently, enable us to draw comparisons on the way the students approached their problem from an affective point of view.
You may provide alternatives in your proposal.
Alternative methods of research may take place in the form of questionnaires or surveys where questions are based on mathematical learning and problem-solving, and beliefs about self in relation to mathematics. Other tools such as interviews, could involve questions ascertaining this kind of conviction to paint a more accurate picture of what students believe about the discipline of mathematics.
Conclude with some explanation as to why this research has a potential to generate useful knowledge for classroom practice.
It aims to reinforce that affective variables are vital in influencing the learning environment in a classroom. Students' willingness to work on a variety of mathematics tasks and their persistence in dealing with these tasks might make a difference in their mathematical performance.
- From the Princess Elizabeth Primary School problem solving video, it was observed that the teacher regularly encouraged student participation and allowed students the freedom of expression. Even when students came up with the wrong concepts or incorrect answers, she did not put them down but offered them other alternatives and various process pathways to generate further thinking.
- The majority of the students displayed good enthusiasm and during the heterogeneous small-group discussions, they appeared to have good constructive interactions among themselves.
- It was also noted that when students did not obtain the correct answers to the questions posed by the teachers, they do not appear to be discouraged and yet showed perseverance to come up with alternative ideas to problem solve.
Your research proposal should include a statement of what you feel is productive to research further into.
For the lesson, it is essential that students have prior cognitive knowledge on the basic mathematical operations. However, as students go about their tasks, it was not easy to evaluate the difficulties and emotions experienced by the individual students and how they eventually react to any stalemates or deadlocks. Other than the cognitive domain, the affective traits could have imposed an influence on the development of the students’ behaviour and thinking. For example, students who were not participative may have believed that if they could not solve a problem in a few minutes, they would never find the solution and thus should not waste time on it. Such beliefs about mathematics shape students’ behaviour and can often produce negative consequences. Thus, it is imperative that classroom teachers know what the different affective traits are, how they influence the students’ success in mathematical problem solving; and finally, how they can nurture the desired affective traits in the students. It was hoped that from the research, the role of affective traits in mathematical problem solving can be better understood, particularly in the field of proving mathematical statements.
Outline how you would go about to do this research.
Case studies will be done on two students within a Secondary 3 Express class of above-average ability. The class will be tasked to prove a particular mathematical statement and to indicate their proving procedures and the thoughts that accompanied the processes. The goal of this mathematical task is to make the students reflect on what happens when a proof is carried out and to develop skills in analyzing during proving processes. The written protocols of two students will then be examined.
These two case studies aim to illustrate that when it comes to mathematical problem solving, purely cognitive behaviour is extremely rare and that the cognitive and affective domains are intertwined. The expressions and quotes presented by the students will allow us to go deeper into the difficulties faced by the two students in terms of the lack of mathematical knowledge and skills and subsequently, enable us to draw comparisons on the way the students approached their problem from an affective point of view.
You may provide alternatives in your proposal.
Alternative methods of research may take place in the form of questionnaires or surveys where questions are based on mathematical learning and problem-solving, and beliefs about self in relation to mathematics. Other tools such as interviews, could involve questions ascertaining this kind of conviction to paint a more accurate picture of what students believe about the discipline of mathematics.
Conclude with some explanation as to why this research has a potential to generate useful knowledge for classroom practice.
It aims to reinforce that affective variables are vital in influencing the learning environment in a classroom. Students' willingness to work on a variety of mathematics tasks and their persistence in dealing with these tasks might make a difference in their mathematical performance.
Teachers have a remarkable influence on students’ construction of their beliefs through the ways in which they present their subject matter, the kinds of tasks they set, assessment methods, procedures and criteria. (Pehkonen, 1998; Pehkonen & To¨rner, 1996; To¨rner, 1998). Thus, it is hoped that the research will encourage educators to acquire better ways to deal with affective issues in mathematical problem solving. This will help to motivate the students by stimulating their affective response in various situations and giving them a sense of better understanding of the subject matter. This would then contribute towards assisting students to construct more advanced and productive beliefs about mathematical problem solving.Demonstrate how you can use a tool such as the blog to promote research in mathematical problem solving.
The blog also allows us to interact with one another and to share opinions and feedback on research findings. Blogs offer us the option of enabling a "comments" field after our posts where readers can give us feedback. This not only encourages our fellow teachers to feel more inclined to revisit the blog to view the responses and to keep ourselves updated about any new research findings, but also at our disposal an effective and inexpensive way to get to know one another better.
References
Pehkonen, E., & To¨ rner, G. (1996). Mathematical beliefs and different aspects of their meaning. International Review on Mathematical Education (ZDM), 28, 101–108.
Pehkonen, E. (1998). Teachers’ conceptions on mathematics teaching. In M. Hannula (Ed.), Current state of research on mathematical beliefs V (pp. 58–65). Proceedings of the MAVI-5 workshop Helsinki, Finland: Department of Teacher Education, University of Helsinki.
To¨rner, G. (1998). Mathematical beliefs and their impact on the students’ mathematical performance. Questions raised by the TIMSS results. In M. Hannula (Ed.), Current state of research on mathematical beliefs V (pp. 83–91). Proceedings of the MAVI-5 workshop. Helsinki, Finland: Department of Teacher Education, University of Helsinki.
Pehkonen, E., & To¨ rner, G. (1996). Mathematical beliefs and different aspects of their meaning. International Review on Mathematical Education (ZDM), 28, 101–108.
Pehkonen, E. (1998). Teachers’ conceptions on mathematics teaching. In M. Hannula (Ed.), Current state of research on mathematical beliefs V (pp. 58–65). Proceedings of the MAVI-5 workshop Helsinki, Finland: Department of Teacher Education, University of Helsinki.
To¨rner, G. (1998). Mathematical beliefs and their impact on the students’ mathematical performance. Questions raised by the TIMSS results. In M. Hannula (Ed.), Current state of research on mathematical beliefs V (pp. 83–91). Proceedings of the MAVI-5 workshop. Helsinki, Finland: Department of Teacher Education, University of Helsinki.
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