Yeo, Joseph B. W.

This paper reports on the types of problems that high-achieving students posed when given investigative tasks that were constructed by opening up some mathematical problems. A teaching experiment was conducted to develop the students’ thinking processes during mathematical investigation, and each student was videotaped thinking aloud during a pretest and a posttest. The findings show that some students were unable to pose the original intended problems and what Krutetskii (1976) called problems that ‘naturally follow’ from the task, including extending the task to generalise. The implications of the difficulty encountered by these students for teaching and research will also be discussed.

This paper introduces a new framework to model the interactions of the processes of specialising and conjecturing when students engage in mathematical investigation. The framework posits that there is usually a cyclic pathway alternating between examining specific examples (specialising) and searching for pattern (conjecturing), instead of a linear pathway as in many other theoretical models. The framework also distinguishes between observing a pattern and formulating it as a conjecture, unlike most models that treat an observed pattern as a conjecture to be proven or refuted. I will then use the framework to analyse and explicate a secondary school student's specialising and conjecturing processes while he attempted an open investigative task.

Many educators believe that mathematical investigation is open and it involves both problem posing and problem solving, but some teachers have taught their students to investigate during problem solving. The confusion about the relationship between investigation and problem solving may affect how teachers teach their students and how researchers conduct their research. Moreover, there is a research gap in studying the thinking processes in mathematical investigation, partly because it is not easy to define these processes. Therefore, this article seeks to address these issues by first distinguishing between investigation as a task, a process and an activity; and then providing an alternative characterisation of the process of investigation in terms of its core cognitive processes: specialising, conjecturing, justifying and generalising. These will help to clarify the relationship between investigation and problem solving: an open investigative activity involves both problem posing and problem solving; but the problem-solving process entails solving by the process of investigation and/or by using "other heuristics". In other words, mathematical investigation does not have to be open. The article concludes with some implications of this alternative view of mathematical investigation on teaching and research.

Joseph B. W. Yeo describes an interactive algebra computer system to help students explore algebra and calculus.

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.

This article presents an exploratory study to find out whether high-ability secondary school students in Singapore were able to deal with open mathematical investigative tasks. A class of Secondary One (or Grade 7) students, who had no prior experience with this kind of investigation, were given a paper-and-pencil test consisting of four open tasks. The results show that these students did not even know how to begin, despite sample questions being given in the first two tasks to guide and help them pose their own problems. The main difficulty was the inability to understand the task requirement: what does it mean to investigate? Another issue was the difference between searching for any patterns without a specific problem to solve, and searching for patterns to solve a given problem. The implications of these findings on teaching and on research methodologies that rely on paper-and-pencil test instruments will also be discussed.