Characterising the cognitive processes in mathematical investigation
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.
Investigating the processes of mathematical investigation
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.
Mathematical investigation: Task, process and activity
Many writers believe that mathematical investigation is open and it involves both problem posing and problem solving. However, some teachers feel that there is a sense of doing some sort of investigation when solving problems with a closed goal and answer but they are unable to identify the characteristics of this type of investigation. Such confusion will affect how teachers teach their students and how researchers conduct their research on investigation. Therefore, this article seeks to clarify the relationship between investigation and problem solving by providing an alternative characterisation of mathematical investigation as a process involving specialisation, conjecturing, justification and generalisation. It also distinguishes between mathematical investigation as a process and as an activity: investigation, as a process, can occur when solving problems with a closed goal and answer, while investigation, as an activity involving open investigative tasks, includes both problem posing and problem solving. Implicit support for this alternative characterisation of mathematical investigation is gathered from some existing literature as these writers did not state this perspective explicitly. The article concludes with some implications of this alternative view on teaching and research.