Mathematical investigation refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended.
I first heard about math investigations in 1990 when I attended a postgraduate course in Australia. I love it right away and it has since become one of my favorite mathematical activity for my students who were so proud of themselves when they finished their first investigation.
Problem solving is a convergent activity. It has definite goal – the solution of the problem. Mathematical investigation on the other hand is more of a divergent activity. In mathematical investigations, students are expected to pose their own problems after initial exploration of the mathematical situation. The exploration of the situation, the formulation of problems and its solution give opportunity for the development of independent mathematical thinking and in engaging in mathematical processes such as organizing and recording data, pattern searching, conjecturing, inferring, justifying and explaining conjectures and generalizations. It is these thinking processes which enable an individual to learn more mathematics, apply mathematics in other discipline and in everyday situation and to solve mathematical (and non-mathematical) problems.
Teaching through mathematical investigation allows for students to learn about mathematics, especially the nature of mathematical activity and thinking. It also make them realize that learning mathematics involves intuition, systematic exploration, conjecturing and reasoning, etc and not about memorizing and following existing procedures. The ultimate aim of mathematical investigation is develop students’ mathematical habits of mind.
Although students may do the same mathematical investigation, it is not expected that all of them will consider the same problem from a particular starting point. The “open-endedness” of many investigation also means that students may not completely cover the entire situation. However, at least for a student’s own satisfaction, the achievement of some specific results for an investigation is desirable. What is essential is that the students will experience the following mathematical processes which are the emphasis of mathematical investigation:
- systematic exploration of the given situation
- formulating problems and conjectures
- attempting to provide mathematical justifications for the conjectures.
In this kind of activity and teaching, students are given more opportunity to direct their own learning experiences. Note that a problem solving task can be turned into an investigation task by extending the problem by varying for example one of the conditions. To know more about problem solving and how they differ with math investigation read my post on Exercises, Problem Solving and Math Investigation.
Some parents and even teachers complain that students are not learning mathematics in this kind of activity. Indeed they won’t if the teacher will not discuss the results of the investigation, highlight and correct the misconceptions, synthesize students’ findings and help students make connection among the math concepts covered in the investigation. This goes without saying that teachers should try the investigation first before giving it to the students.
I think mathematical investigation is constructivist teaching at its finest. For a sample lesson, read Polygons and algebraic expressions.