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The nature and development of processes in mathematical investigation
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Type
Thesis
Abstract
The purpose of this research study is to examine the nature and development of cognitive and metacognitive processes that students use when attempting open investigative tasks. Mathematical investigation is important in many school curricula because many educators think that school students should do some real mathematics, the mathematics which academic mathematicians do in their daily and working lives, investigating and solving problems to discover new mathematics. They believe in the benefits of the processes that these mathematicians engage in, e.g. problem posing, specialising, conjecturing, justifying and generalising. Thus it is vital to understand the nature of these processes (i.e. the types of investigation processes and how they interact with one another), and how they can be developed, so that the teachers are better informed to cultivate these processes in their students. Currently, there is a research gap in this field, as there are few empirical studies on processes in mathematical investigation. Therefore, this research study could add value to the advancement of mathematics education in this area.
The sample for the main study consisted of 10 Secondary Two (equivalent to Grade 8) students from a high- performing Singapore school. They went through a teaching experiment consisting of a familiarisation lesson and five developing lessons. The duration of each lesson was two hours. They sat for a pretest at the end of the familiarisation lesson, and a posttest at the end of the last developing lesson. Each student was separately videotaped thinking aloud while working on two open investigative tasks (one from Type A and the other one from Type B) in each test. The verbal protocols were transcribed and coded using a coding scheme, which had passed an inter-coder reliability test. The coded transcripts were then analysed qualitatively to validate and refine the two theoretical investigation models for cognitive and metacognitive processes formulated for this research, to study the effect of these processes on the investigation outcomes, and to examine the development of these processes. A scoring rubric was also devised to score the pretest and the posttest in order to study the effect of the teaching experiment on the development of the investigation processes quantitatively using descriptive statistics.
The findings indicated that the two types of investigative tasks tend to elicit different types of investigation processes and investigation pathways: for Type A, students set out to search for any pattern by specialising, conjecturing, justifying and generalising; for Type B, students posed specific problems to solve by using other heuristics, such as reasoning, and then they extended the task by changing the given in order to generalise. Some new cognitive and metacognitive processes and outcomes were also found, which resulted in the refinement of the two theoretical investigation models. Data analysis showed that there was no direct relationship between the completion of an investigation pathway and the types of investigation outcomes produced. The study also identified the processes that had helped the students to produce significant or non-trivial outcomes in their investigation, the processes that were developed more fully in the students during the teaching experiment, and the processes that were still lacking in the students. The implication was that it is possible to develop investigation processes by teaching the students these processes and providing them the opportunity to develop these processes when they attempt suitable investigative tasks. The research also revealed which processes took a longer time to develop, so more attention should be paid to cultivate these processes during teaching.
The sample for the main study consisted of 10 Secondary Two (equivalent to Grade 8) students from a high- performing Singapore school. They went through a teaching experiment consisting of a familiarisation lesson and five developing lessons. The duration of each lesson was two hours. They sat for a pretest at the end of the familiarisation lesson, and a posttest at the end of the last developing lesson. Each student was separately videotaped thinking aloud while working on two open investigative tasks (one from Type A and the other one from Type B) in each test. The verbal protocols were transcribed and coded using a coding scheme, which had passed an inter-coder reliability test. The coded transcripts were then analysed qualitatively to validate and refine the two theoretical investigation models for cognitive and metacognitive processes formulated for this research, to study the effect of these processes on the investigation outcomes, and to examine the development of these processes. A scoring rubric was also devised to score the pretest and the posttest in order to study the effect of the teaching experiment on the development of the investigation processes quantitatively using descriptive statistics.
The findings indicated that the two types of investigative tasks tend to elicit different types of investigation processes and investigation pathways: for Type A, students set out to search for any pattern by specialising, conjecturing, justifying and generalising; for Type B, students posed specific problems to solve by using other heuristics, such as reasoning, and then they extended the task by changing the given in order to generalise. Some new cognitive and metacognitive processes and outcomes were also found, which resulted in the refinement of the two theoretical investigation models. Data analysis showed that there was no direct relationship between the completion of an investigation pathway and the types of investigation outcomes produced. The study also identified the processes that had helped the students to produce significant or non-trivial outcomes in their investigation, the processes that were developed more fully in the students during the teaching experiment, and the processes that were still lacking in the students. The implication was that it is possible to develop investigation processes by teaching the students these processes and providing them the opportunity to develop these processes when they attempt suitable investigative tasks. The research also revealed which processes took a longer time to develop, so more attention should be paid to cultivate these processes during teaching.
Date Issued
2013
Call Number
QA14.S55 Yeo
Date Submitted
2013