Yeo Boon Wooi Joseph

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.

In this paper, the differences between mathematical tasks such as problem-solving tasks, investigative tasks, guided-discovery tasks, project work, real-life tasks, problem-posing tasks, open tasks and illstructured tasks will be contrasted. Such clarification is important because it can affect how and what teachers teach since the diverse types of tasks have different pedagogical uses, and it can also help researchers to define more clearly the tasks that they are investigating on. A framework to characterise the openness of mathematical tasks based on task variables such as the goal, the method, the answer, scaffolding and extension will be described. The tasks are then classified according to their teaching purpose: mathematically-rich tasks, such as analytical tasks and synthesis tasks, can provide students with opportunities to learn new mathematics and to develop mathematical processes such as problemsolving strategies, analytical thinking, metacognition and creativity; and non-mathematically-rich tasks, such as procedural tasks, can only provide students with practice of procedures. Rich assessment tasks that teachers can use to assess students’ conceptual understanding, mathematical communication and thinking processes will also be discussed. The clarification of terminologies and the classification of mathematical tasks will help teachers to understand more about the purpose and characteristics of the diverse types of tasks so that they can choose appropriate tasks to develop the different facets of their students’ mental structures and to assess the various aspects of their learning.

Many teachers have experienced at one time or another the frustrations of trying to impart their knowledge to their students but the latter somehow seem not to grasp the full meaning of the content taught. This may be due to the constructivists’ belief that knowledge cannot be transmitted from teachers to learners but is actively constructed by the learners themselves as they attempt to make sense of their experiences. So this paper attempts to look at how mathematics teachers can redesign their pedagogy by taking into account new teaching methods that are made possible by technology. The paper will also give a few examples of how to use various mathematical software to guide pupils to explore mathematical concepts so that they can construct their own knowledge.

The recent revision of the Singapore secondary mathematics syllabuses emphasises seeing mathematics as a tool and as a discipline. Doing this requires teachers to design and implement inquiry-based learning activities with their students. However, it is not always clear what inquiry-based learning entails and what it means for students to learn mathematics as a discipline. In this paper, we discuss what it means to think like a mathematician and illustrate three forms of inquiry-based teaching approaches with some examples for teachers to consider.

The effect of computer-based learning using LiveMath, an interactive computer algebra system, on Singapore Secondary Four students' conceptual and procedural knowledge of exponential and logarithmic curves, was investigated in this study. Sixty-five students from two middle-ability Express classes in an independent school were taught using a guided discovery approach to explore the characteristics of the graphs of exponential and logarithmic hnctions. The experimental class used LiveMath for their exploration whereas the control class used worksheets that contained pre-printed graphs. The findings show that the students in the experimental group performed significantly better than the students in the control group in both the conceptual and the procedural knowledge tests.

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.

This paper describes a research study on how and what secondary school students investigate when faced with an open investigative task involving an interesting game that combines magic square and tic-tac-toe. It will examine the strategies that the students use and the mathematical thinking processes that they engage in when doing their investigation. The findings will be used to inform a theoretical model that we have devised to study the cognitive processes of open mathematical investigation, which include understanding the task, posing problems to investigate, specialising, formulating and testing conjectures, generalising, looking back and extending the task.