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 anchored on 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.

The books below offers investigation “start-up”. The first book is titled “Problems for Student Investigation: Resources for Calculus” is a for college students. The second shows examples of real life investigation.

To me, Math investigation is an ally of constructivism. Students are engaged in tasks that allow them to put in more on what they know as they tinker on a math problem. They do exploration and research to be able to shed more light to the problem. Creative and critical thinking come to the fore. The investigation may take time. But, at the end of the process, students defend what they got, defend it, and share it.

Math investigation takes time. It is quality time. The teacher stands by for consultation and extends assistance, as needed by the students.

My book “High School Geometry with Practical Work and Portfolio Making” is rich on math investigations. My other book “Science Research and Statistics” has much more on investigations particularly on Science Investigatory projects.

hello!

i am trying to teach my students to appreciate the process of mathematical investigation and to develop the skills needed. however, i am having a hard time organizing my lessons. i want to them to develop the thinking skills necessary for them to create their own math IPs or math research papers later on.

1. do you have any suggestions or activities or lesson plans that i can refer to so that the skills can be developed gradually?

2.can you recommend any textbook that can be used for this purpose alone including different mathematical model?

i hope that you could answer my queries.

thank you very much.

Ms. Daryl

Hi Daryl,

It’s best to expose students to problems with many solutions/answers first before giving them mathematical investigation tasks. You can also convert some of your exercises to problems and mathematical investigation. Check out my post on exercises, problems and math investigations.

erronda

I think our use of the Connected Math series really falls in line with this post. Curriclum planning needs to become as strignent as lessons planning that we as teachers work extremely hard on.